Properly colored cycles in edge-colored graphs

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P

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OLORED

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PROPERLY COLORED CYCLES IN

EDGE-COLORED GRAPHS

DISSERTATION

to obtain

the degree of doctor at the University of Twente, on the authority of the rector magnificus,

prof.dr. T.T.M. Palstra,

on account of the decision of the graduation committee, to be publicly defended

on Friday the 2nd of February 2018 at 14.45 hrs

by

Ruonan Li

born on the 14th of August 1991 in Xi’an, China

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The research reported in this thesis has been carried out within the frame-work of the MEMORANDUM OF AGREEMENT FOR A DOUBLE DOCTORATE DEGREE BETWEEN NORTHWESTERN POLYTECHNICAL UNIVERSITY, PEOPLE’S REPUBLIC OF CHINA AND THE UNIVERSITY OF TWENTE, THE NETHERLANDS

IDS Ph.D. Thesis Series No. 18-457 Institute on Digital Society

P.O. Box 217, 7500 AE Enschede, The Netherlands. ISBN: 978-90-365-4471-9

ISSN: 2589-4730 (IDS Ph.D. thesis Series No. 18-457) DOI: 10.3990/1.9789036544719

Available online at

http://doi.org/10.3990/1.9789036544719

Typeset with LA_TEX

Printed by Gildeprint Cover design by Ruonan Li

Copyright c_{2018 Ruonan Li, Enschede, The Netherlands}

All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, me-chanical, photocopying, recording, or otherwise, without prior permission from the copyright owner.

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Graduation Committee Chairman/secretary: prof. dr. P.M.G. Apers Supervisors:

prof. dr. ir. H.J. Broersma prof. dr. S. Zhang Members: prof. dr. M.J. Uetz prof. dr. J.L. Hurink prof. dr. G.Z. Gutin prof. dr. I. Schiermeyer University of Twente University of Twente

Northwestern Polytechnical University

University of Twente University of Twente

Royal Holloway College, London

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Preface

This thesis is the result of four years of research in the area of edge-colored graphs. The first paper in this field that I read was “Rainbow triangles in edge-colored graphs” by Li et al. [38]. I clearly remember how excited I was when I noticed that the extremal cases in their paper correspond to directed graphs. They proved the existence of directed triangles in directed graphs (under a certain condition), and then fed back their result to the existence problem for rainbow triangles in edge-colored graphs. At that time, I started wondering about the relationship between edge-colored graphs and directed graphs with respect to other cycle-related problems. Studying this relationship has been the main motivation for the work and results reported in this thesis.

The chapters in this thesis are mostly based on papers and reports written during the last four years. Apart from the introductory chapter, all chapters have the structure of a journal paper. In the introductory chapter, we have at-tempted to make the terminology and notation used in the subsequent chap-ters consistent. There are several more specific terms and notations that are not defined in the introductory chapter, but they can be found in the chapters where they are used.

The first three chapters (Chapters 2, 3 and 4 ) deal with the existence of short or long properly edge-colored cycles in different types of edge-colored graphs. These chapters are mainly based on the research that the author carried out while she was working as a PhD student in Northwestern Poly-technical University in Xi’an, China. The final three chapters, that were moti-vated by the relationship between edge-colored graphs and directed graphs,

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have been completed when the author was a visiting joint PhD student at the University of Twente.

Papers underlying this thesis

[1] Color degree sum conditions for rainbow triangles in edge-colored graphs, Graphs and Combinatorics 32 (2016), 2001–2008 (with B. Ning and S.

Zhang). (Chapter 2)

[2] Color degree and monochromatic degree conditions for short properly col-ored cycles, Journal of Graph Theory, 87 (2018), 362–373. (with S. Fujita

and S. Zhang). (Chapters 2 and 3)

[3] Cycle extension in edge-colored complete graphs, Discrete Mathematics 340 (2017), 1235–1241 (with H.J. Broersma, C. Xu and S. Zhang). (Chapter 4) [4] Vertex-disjoint properly edge-colored cycles in edge-colored complete

graphs, submitted (with H.J. Broersma and S. Zhang). (Chapter 5)

[5] Decomposing edge-colored graphs under color degree constraints, Electronic Notes in Discrete Mathematics 61 (2017), 491–497 (with S. Fujita and G.

Wang) . (Chapter 6)

[6] Properly edge-colored theta graphs in edge-colored complete graphs,

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Introduction

Graph theory is often used to study relations between pairs of members of a given set. In such cases, the members and the existing relations between pairs of members are usually modeled as vertices and edges (or arcs), respec-tively. For undirected graphs and each (unordered) pair of distinct vertices uand v, there are two possible relations, namely that u and v are adjacent (indicated by an edge uv= vu joining the two vertices) or nonadjacent (no edge between the vertices). For directed graphs and each ordered pair of distinct vertices u and v, there are four possible relations, i.e., dominating only (an arc uv indicates that u dominates v), dominated only (an arc vu indicates that u is dominated by v), dominating and dominated (both arcs are present), and nonadjacent (none of the arcs is present). If more possi-bilities for relations exist and have to be modelled, then neither undirected graphs nor directed graphs are suitable for constructing a proper mathemat-ical model. However, by representing each of the possible relations by an edge (or arc) with a unique color, one can characterize the total situation of all the pairwise relations by an edge-colored (arc-colored directed) graph. Note that thus far, we have not put any restrictions on the edge-coloring, so here an edge-coloring is just a mapping from the edge set of a graph to a set of colors in which each edge is assigned one of the colors. In particular, ad-jacent edges, i.e., edges that share precisely one vertex, are allowed to have the same color.

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Research related to edge-colored graphs can be generally classified into two classes. One class consists of research in which the purpose is to assign the colors to the edges of a graph with certain requirements or restrictions. A typical example is determining the edge-chromatic number [6], i.e., de-ciding the minimum number of colors that is needed to color the edges of a graph such that each pair of adjacent edges are assigned different colors. The other class consists of research in which the goal is to show the existence of (or find) subgraphs with certain coloring characteristics in a graph that has already been edge-colored. Typical examples here are Ramsey theory[52], dealing with the existence of monochromatic cliques in edge-colored com-plete graphs, anti-Ramsey theory [30], dealing with the existence of hete-rochromatic subgraphs in edge-colored complete graphs, and the rainbow Turán problem[23, 35], dealing with the existence of heterochromatic sub-graphs in properly edge-colored sub-graphs. Most of the research reported in this thesis, dealing with the existence of special types of cycles –properly edge-colored cyclesin edge-colored graphs, belongs to the second class.

Properly edge-colored cycles (or PC cycles for short) are edge-colored cy-cles in which adjacent edges are assigned different colors. Bang-Jensen and Gutin suggested in [11] that the concept of PC cycles has most likely first appeared in a paper of Petersen, dating back to 1891[51] (cf. [49]). This paper contains a paragraph with the following content: if the edges of an α-regular graph G can be partitioned into a spanning β-regular graph H1

and a spanning γ-regular graph H2 withα = β + γ, and G contains a cycle

C with its edges alternating between H₁ and H₂, then we can obtain a span-ningβ-regular graph H₁0 and a spanningγ-regular graph H0₂of G, which are different from H1 and H2, respectively. The proof for this statement is easy:

when we assign color red to all edges of H₁, and color blue to all edges of H2, then swapping the colors on the edges along the cycle C, we obtain the

required spanning subgraphs H₁0 and H0₂. PC cycles have also appeared in applications, e.g., in social science[21] and molecular biology [25]. Here, we are interested in theoretical aspects only.

Finding PC cycles in edge-colored graphs or proving their existence is in-tuitively more difficult than finding cycles or proving their existence in graphs and directed graphs. As an example, consider edge-colored complete graphs

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1.1. Terminology and notation 3

versus tournaments (i.e., oriented complete graphs). For the existence of short directed cycles, it is easy to show that a tournament containing a di-rected cycle of length` also contains directed cycles of all lengths between 3 and`. However, an edge-colored complete graph containing a PC cycle may not contain a PC triangle. For the existence of long cycles, it is known that if a tournament contains a directed cycle C of length` and a vertex v not on C such that C contains both in-neighbors and out-neighbors of v, then we can add v to C and obtain a directed cycle of length ` + 1. However, for a PC cycle C of length` in an edge-colored complete graph and a vertex v not on C, even when v is joined to C with edges of` different colors, it is possible that v is not contained in any PC cycles.

In spite of the above observations, there are some intuitive properties that one could use for showing the existence of PC cycles. For example, if a rainbow cycle C (i.e., a cycle in which all edges have different colors) has a chord uv, then either uC+vuor uC−vu(where uC+v and uC−v denote the two distinct paths on C from u to v) is a shorter rainbow cycle. For a PC cycle C, a vertex v not on C and a vertex u on C which is adjacent to v, the color of uv must be distinct to one of the colors on uu+and uu−(where u+and u− denote, respectively, the immediate successor and predecessor of u on C in an arbitrary fixed orientation along C). If the color of vu is distinct to both colors on uu+ and uu−, and there are at least two colors appearing on the edges between v and C, then v is also contained in a PC cycle.

The reader may see in later chapters that many lemmas and proof tech-niques are generated from these intuitive properties.

1.1 Terminology and notation

All graphs considered in this thesis are finite, simple, and undirected unless specified explicitly as directed graphs. For terminology and notation not de-fined here, we refer the reader to[18].

For a, b_{∈ N with a ≤ b, we use [a, b] to denote {i ∈ N : a ≤ i ≤ b}.} Let G be a graph. We use V(G) and E(G) to denote the set of vertices and edges of G, respectively. For a vertex v _{∈ V (G), denote by N}_G(v) the

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neighborhoodof v, i.e., the set of all vertices adjacent to v, and by d_G(v) = |NG(v)| the degree of v in G. For two disjoint subsets A and B of V (G), denote

by E_G(A, B) the set of edges in G between A and B. If A contains only one vertex v, we write E_G(v, B) instead of E_G({v}, B). For two vertex-disjoint subgraphs F and H of G, we use E_G(H, F) to denote E_G(V (H), V (F)). When there is no ambiguity, we often write N(v) for N_G(v), d(v) for d_G(v), and E(A, B) for E_G(A, B). For a nonempty subset X ⊆ V (G), we use G[X ] to denote the subgraph of G induced by X , i.e., the subgraph on vertex set X containing all the edges of G between pairs of vertices in X . A path (of length k) in G is a sequence of distinct vertices v1, v2, . . . , vk+1 such that vivi+1 ∈ E(G) for all i ∈ {1, 2, . . . , k}. A cycle (of length k ≥ 3) consists of a path (of length k_{− 1) together with an edge joining the two end-vertices of the} path. A Hamilton cycle of a graph G is a cycle containing all vertices of G, i.e., a cycle of length_{|V (G)|. A chord of a cycle C is an edge uv between two} distinct vertices of C that are nonadjacent on C.

An edge-coloring of G is a mapping col : E(G) → N, where N is the set of natural numbers. A graph G with an assigned edge-coloring is called an edge-colored graph (or throughout the later chapters of this thesis simply a colored graph). We say that G is a properly edge-colored graph (or PC graph for short) if each pair of adjacent edges of G are assigned distinct colors, and we say that G is a rainbow graph if all edges of G are assigned distinct colors. Let G be an edge-colored graph. For an edge e _{∈ E(G), we use col(e)} to denote the color of e. For a subgraph H of G, we denote by col(H) the set of colors that are appearing on E(H). The cardinality of col(G) is called the color number of G. For vertex-disjoint subgraphs F and H of G, we use col(F, H) to denote the set of colors appearing on the edges between F and H. For a color i _{∈ col(G), we use G}i to denote the subgraph of G induced by _{{e ∈ E(G) : col(e) = i}. We say a color appears (at least k times) at} a vertex v _{∈ V (G) if it is assigned to at least one (at least k) of the edges} incident with v. For a vertex v of G, the color neighborhood of v, denoted by N_Gc(v), is the set of colors appearing on the edges incident to v. The color degreeof v, denoted by d_Gc(v), is the cardinality of N_Gc(v). For a subset S of V(G), we define the minimum color degree of the vertices in S by δc_G(S) = min{dc

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1.2. PC tours and PC cycles 5

in S by∆mon

G (S) = max{dGi(v) : i ∈ col(G), v ∈ S}. For a subgraph H of G,

we defineδ_Gc(H) = δ_Gc(V (H)) and ∆mon_G (H) = ∆mon_G (V (H)). If there is no ambiguity, we often write Nc(v) for N_Gc(v), dc(v) for d_Gc(v), δc(G) for δ_Gc(G), and∆mon_{(G) for ∆}mon

G (G).

Directed graphs are sometimes used as tools in proofs. Let D be a directed graph. We use V(D) and A(D) to denote the sets of vertices and arcs of D, respectively. For a vertex v∈ V (D), we denote by dD+(v) the out-degree of v

in D, i.e., d+_D(v) = |{u ∈ V (D) : vu ∈ A(D)}|, and by d_D−(v) the in-degree of v in D, i.e., d−_D(v) = |{u ∈ V (D) : uv ∈ A(D)}|.

1.2 PC tours and PC cycles

By several authors of standard text books in graph theory, the beginning of graph theory is marked by Euler’s treatment of the problem of the Seven Bridges of Königsberg. In this problem, the aim was to find a round trip, i.e., a walking tour, passing all the seven bridges of Königsberg exactly once. Euler showed that there cannot exist such a tour, by using a graph to model the problem, and by showing that a connected graph G admits such a closed tour traversing each edge exactly once if and only if d(v) is even for each vertex v_{∈ V (G). More generally, it is easy to show that a connected directed} graph D has a closed tour traversing each arc exactly once if and only if d+(v) = d−(v) for each vertex v ∈ V (D). In [37], Kotzig characterized edge-colored graphs containing PC tours, i.e., closed tours in which consecutive edges have different colors, as follows.

Theorem 1.1 (Kotzig[37]). An edge-colored graph G contains a PC tour if and only if G is connected, each vertex of G has even degree, and for every vertex x and every color i∈ col(G), dGi(v) ≤ P_j

6=idGj(v).

From the perspective of the existence of tours of the above types, we ob-serve a striking similarity among graphs, digraphs and edge-colored graphs. However, in contrast, when we consider cycle-related problems, the situation is quite different, as we will see.

Cycles play an important role in graph theory and its applications. We know that every graph G with δ(G) ≥ 2 contains a cycle, and that every

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directed graph D with δ+(D) ≥ 1 contains a directed cycle. However, for any constant k ∈ N, there exists an edge-colored graph G with δc_{(G) ≥ k}

containing no PC cycles. See the graph in Figure 1.1 given by Wang and Li[61] for an example.

(a) G1

H1 H2 H3 · · · Hi+1

vi+1

(b) Gi+1

FIGURE1.1: Each H_j(1≤ j ≤ i + 1) is a copy of Gi

The following theorem, that is frequently used in later chapters, charac-terizes edge-colored graphs containing no PC cycles.

Theorem 1.2 (Grossman and Häggkvist[33], Yeo [65]). Let G be an edge-colored graph containing no PC cycles. Then there is a vertex z ∈ V (G) such that no component of G− z is joined to z with edges of more than one color.

In Chapter 3, we give a minimum color degree condition for the existence of PC cycles in edge-colored graphs. We recommend Chapter 16 in[12] for a survey on PC cycles.

1.2.1 Short PC cycles

We first present some results on shortest PC cycles in edge-colored graphs that are in fact PC triangles, more commonly known as rainbow triangles. In 2012, H. Li and G. Wang[41] conjectured that every edge-colored graph of order n with minimum color degree at least n+1₂ contains a rainbow triangle. In 2013, H. Li [39] confirmed this conjecture himself. Independently, B. Li et al.[38] verified this conjecture by proving a stronger result, and they also characterized the corresponding extremal graphs.

Theorem 1.3 (B. Li et al.[38]). Let G be an edge-colored graph of order n ≥ 3. IfP_v_{∈V (G)}dc(v) ≥ n(n+1)₂ , then G contains a rainbow triangle.

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Theorem 1.4 (B. Li et al.[38]). Let G be an edge-colored graph of order n ≥ 3. Ifδc(G) ≥ n₂ and G contains no rainbow triangles, then n is even and G is the complete bipartite graph Kn

2,

n

2, unless G= K4− e or K4 when n= 4.

Motivated by the relationship between Dirac-type degree conditions and Ore-type degree conditions and the existence of long cycles, in Chapter 2, we establish a color degree sum condition for adjacent vertices that guarantees the existence of rainbow triangles, and we also characterize the extremal graphs.

In[38], Li et al. also gave an “|E(G)| + |col(G)|” condition for the exis-tence of rainbow triangles in an edge-colored graph G.

Theorem 1.5 (B. Li et al.[38]). Let G be an edge-colored graph of order n ≥ 3. If_{|E(G)| + |col(G)| ≥} n(n+1)₂ , then G contains a rainbow triangle.

As a generalization of Theorem 1.5, Xu et al. [64] gave an “|E(G)| + |col(G)|” condition for the existence of rainbow cliques in an edge-colored graph G. Recently, Fujita et al.[31] characterized those edge-colored graphs G with_{|E(G)| + |col(G)| =} n(n+1)

2 and containing only one rainbow triangle.

For the existence of rainbow triangles in edge-colored complete graphs, the following partition theorem due to Gallai from 1976 is very powerful and useful.

Theorem 1.6 (Gallai[32]). If an edge-colored complete graph K_ncontains no rainbow triangle, then V(K_n) can be partitioned into at least two parts such that between the parts, there are a total of at most two colors and, between every pair of parts, there is only one color on the edges.

In 2004, by applying the result of Gallai, Gyárfás and Simonyi[34] ob-tained a maximum monochromatic degree condition for the existence of a rainbow triangle in an edge-colored complete graph. In Chapter 2, we give a sharp color degree condition (δc(K_n) > log₂n) for the existence of rainbow triangles in edge-colored complete graphs, supplied with two independent proofs. One of the proof is based on the partition given by Theorem 1.6, while the other is self-contained.

Interestingly, Axenovich et al.[7] proved that an edge-colored complete graph contains a PC cycle of length 4 if the minimum color degree is at least

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3. As we will see in Chapter 3, this bound is the same as that for PC cycles of length 4 in edge-colored complete bipartite graphs.

Next we deal with some results related to shortest PC cycles in edge-colored complete bipartite graphs, in particular with PC cycles of length 4. In Chapter 3, we characterize the structure of edge-colored complete bipartite graphs containing no PC cycles of length 4. Based on this characterization, we give a minimum color degree condition and a maximum monochromatic degree condition for an edge-colored complete bipartite graph to contain a PC cycle of length 4, and for PC cycles of length 4 passing through a given vertex or edge, respectively. For results on rainbow cycles of length 4 in edge-colored graphs, we refer the reader to[19, 39, 50].

In addition, when considering shortest PC cycles in edge-colored plete graphs, edge-colored complete bipartite graphs and edge-colored com-plete multipartite graphs, we have the following observations which are sim-ilar to but distinct from their counterparts in tournaments, bipartite tourna-ments and multipartite tournatourna-ments. Proofs of these three observations are given in Chapters 5, 3 and Appendix A, respectively.

Observation 1.7. Let G be an edge-colored complete graph containing a PC cycle C. Then G[V (C)] contains a PC cycle of length 3 or 4.

Observation 1.8. Let G be an edge-colored complete bipartite graph containing a PC cycle C. Then G[V (C)] contains a PC cycle of length 4 or 6.

Observation 1.9. Let G be an edge-colored complete multipartite graph con-taining a PC cycle C. Then G[V (C)] contains a PC cycle of length ` ∈ {3, 4, 6}.

1.2.2 Long PC cycles

Problems and results on long PC cycles are mostly driven by several open conjectures and their partial solutions. In 1976, Bollobás and Erd˝os [15] conjectured the following.

Conjecture 1.1 (Bollobás and Erd˝os[15]). Each edge-colored complete graph Knwith∆mon(Kn) < bn₂c contains a PC Hamilton cycle.

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In the same paper, they proved that if ∆mon_(K

n) < ₆₉n, then this

edge-colored K_n contains a PC Hamilton cycle. Later, Chen and Daykin[20], and Shearer [53], respectively, improved the bound to ₁₇n and n₇. By a proba-bilistic method, Alon and Gutin [5] improved the bound to (1 −p1

2 − ")n

(" is small enough) for sufficient large n. Recently, Lo [47] confirmed this conjecture asymptotically.

Theorem 1.10 (Lo[47]). For any arbitrarily small real " > 0, there exists an integer n0= n0(") such that every edge-colored complete graph Kn with n≥ n0

and∆mon(K_n) ≤ (1₂− ")n contains a PC Hamilton cycle.

For a sufficient minimum color degree condition in edge-colored com-plete graphs, Fujita and Magnant[28] conjectured the following.

Conjecture 1.2 (Fujita and Magnant[28]). Let G be an edge-colored K_n. If δc_{(G) ≥} n+1

2 , then each vertex of G is contained in a PC cycle of length k, for

each k with 3≤ k ≤ n.

In Chapter 4, we obtain a cycle extension result and show that each ver-tex in an edge-colored K_n with δc(K_n) ≥ n+1₂ is contained in a PC cycle of length at leastδc(K_n). In Section 7.5 of Chapter 7, by studying the exis-tence of PC theta graphs in edge-colored complete graph under the condition thatδc(K_n) ≥ n+1₂ , we obtain that either this edge-colored K_n contains a PC Hamilton cycle or each maximal PC cycle C contains a chord uv such that C and uv form a PC theta graph.

For color degree conditions in general (not necessarily complete) edge-colored graphs, Lo[45] proved that an edge-colored graph G with δc_{(G) ≥ d}

contains either a PC path of length at least 2d or a PC cycle of length at least d+ 1. In the same paper, it is conjectured that each edge-colored graph G withδc(G) ≥ 2n₃ contains a PC Hamilton cycle. Later, in[46], Lo showed the existence of a PC 2-factor under the condition of the above conjecture, and also verified the conjecture asymptotically. For other conjectures and results on PC Hamilton paths and cycles, we refer the reader to[1, 2, 4, 13, 20, 22, 24, 26, 45, 46, 53, 62].

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1.3 PC cycles and directed cycles

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. Let D be a directed graph with vertex set _{v₁, v₂, . . . , v_n_{}. Color each arc e = v}_iv_j with the color c_j and remove the orientations of arcs. Then we obtain an edge-colored graph G (See Figure 1.2 for an example), and PC cycles in G are in one-to-one correspondence with directed cycles in D.

v1 v2

v3

v4

v5

v6

(a) A directed graph D

v1 v2 v3 v4 v5 v6 (b) An edge-colored graph G

FIGURE 1.2: The transition between directed graphs and edge-colored graphs.

Based on this observation, in this sense the research on PC cycles in edge-colored graphs generalizes the research on directed cycles in directed graphs. We recommend the proof techniques in[38] and [46], and the constructions in [28] and Chapter 16 of [12] for further details on the close relationship between edge-colored graphs and directed graphs. The last three chapters in this thesis are motivated by and dedicated to this relationship. In particular, the results in Chapters 5 and 6 are related to the following conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.

Conjecture 1.3 (Bermond and Thomassen[14]). Let D be a directed graph. Ifδ+(D) ≥ 2k − 1, then D contains k vertex-disjoint directed cycles.

In Chapter 5, it is conjectured that every edge-colored complete graph G on n vertices satisfying ∆mon_{(G) ≤ n − 3k + 1 contains k vertex-disjoint}

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1.3. PC cycles and directed cycles 11

weaker results for general k, and establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close rela-tionship between PC cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to the above conjecture.

In Chapter 6, we consider a partitioning problem with respect to the min-imum color degree condition, i.e., the problem of determining the minmin-imum function f(a, b) such that each edge-colored graph G with δc_{(G) ≥ f (a, b)}

can be partitioned into two parts A and B such that δc(G[A]) ≥ a and δc_{(G[B]) ≥ b. We prove that if G is an edge-colored graph with δ}c_{(G) ≥ 5,}

then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least 2. We establish this theorem by proving a much stronger form. Moreover, we point out an important relation-ship between our theorem and the aforementioned Bermond-Thomassen’s conjecture in digraphs.

In Chapter 7, we consider subgraph structures that extend PC cycles – PC theta graphs. A theta graph consists of two vertices with degree 3 and three internally vertex-disjoint paths joining them. For each pair of distinct vertices uand v on a PC cycle, there are two internally-disjoint PC paths from u to v with distinct starting colors and distinct ending colors. For each PC theta graph, there exist two distinct vertices u and v and three internally-disjoint PC paths from u to v with distinct starting colors and distinct ending colors. We show that PC theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. The color number condition for an edge-colored complete graph to contain a PC cycle is that_|col(K_n)| ≥ n. We prove that if |col(K_n)| ≥ n + 1, then this edge-colored complete graph contains a PC theta graph, and that the lower bound on_|col(K_n)| is sharp. We also consider sufficient color degree conditions for the existence of PC theta graphs in edge-colored complete graphs and obtain the following. Ifδc(K_n) ≥ n+1₂ , then one of the following statements holds: (i) dc_{(u) =} n+1

2 for each vertex u∈ V (Kn) and this edge-colored Kn contains

a PC Hamilton cycle; (ii) each maximal PC cycle C in this edge-colored K_n has a chord uv such that{uv, uC+v, uC−v} is a PC theta graph. This result is

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Chapter 2

Rainbow triangles in

edge-colored graphs

In this chapter, we first study the existence of rainbow triangles in edge-colored (general) graphs and then give a sharp color degree condition for the existence of rainbow triangles in edge-colored complete graphs. As we mentioned in the previous chapter, we will use the term colored as shorthand for edge-colored throughout this chapter.

2.1 Introduction

Rainbow subgraphs in colored graphs, such as rainbow matchings and rain-bow cycles etc., have been well studied (See the survey paper [36]). Here we mainly focus on the existence of rainbow triangles in colored graphs.

Let G be a graph on n vertices. We know from Mantel’s theorem that G contains a triangle if_{|E(G)| > bn}2/4c. As a corollary, G contains a triangle if d(v) ≥ (n + 1)/2 for every vertex v ∈ V (G).

For a colored graph G, Li and Wang [40] conjectured in 2006 that G contains a rainbow triangle if dc(v) ≥ (n + 1)/2 for every vertex v ∈ V (G). This conjecture was formally published in [41] in 2012 and confirmed by Li[39] in 2013.

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Theorem 2.1 (Li [39]). Let G be a colored graph on n vertices. If dc_{(v) ≥}

(n + 1)/2 for every vertex v ∈ V (G), then G contains a rainbow triangle. Independently, Li et al. [38] proved a stronger result, obtaining Theo-rem 2.1 as a corollary.

Theorem 2.2 (Li et al.[38]). Let G be a colored graph on n vertices. If X

v∈V (G)

dc(v) ≥ n(n + 1)/2,

then G contains a rainbow triangle.

Li et al.[38] also proved that the bound of color degree in Theorem 2.1 is tight for the existence of rainbow triangles, but can be lowered to n/2 with some simple exceptions.

Theorem 2.3 (Li et al.[38]). Let G be a colored graph on n vertices. If dc(v) ≥ n/2 for every vertex v ∈ V (G) and G contains no rainbow triangles, then n is even and G is a PC K_n/2,n/2, unless G= K4− e or K4 when n= 4.

Motivated by the relation between the classic Dirac’s condition and Ore’s condition for long cycles, we wonder whether a colored graph G on n vertices contains a rainbow triangle when

dc(u) + dc(v) ≥ n + 1 (2.1)

for every nonadjacent vertices u, v_{∈ V (G).}

In fact, Bondy [16] proved that a graph G on n vertices is pancyclic if d(u) + d(v) ≥ n + 1 for any nonadjacent vertices u, v ∈ V (G). Certainly, G contains a triangle when G is pancyclic.

However, we find that the color degree sum condition (2.1) can not guar-antee the existence of rainbow triangles.

Example 2.1. Construct a colored graph G with V(G) = {v₁, v₂, . . . , v_n_}, E(G) = {v_iv_j: 1≤ i < j ≤ n, 1 ≤ i ≤ dc/2e}, and col(vivj) = min{i, j}, where c ∈ [n + 1, 2n − 2] is an integer. Obviously, G satisfies that dc(u) + dc(v) ≥ c ≥ n + 1 for every pair of nonadjacent vertices u, v ∈ V (G) but contains no rainbow triangles.

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2.2. Two lemmas 15

Oppositely, motivated by the fact that a graph G contains a triangle if there is an edge uv∈ E(G) satisfying d(u) + d(v) ≥ |V (G)| + 1, we show that the color degree sum condition for adjacent vertices is able to guarantee the existence of rainbow triangles in colored graphs.

Theorem 2.4. Let G be a colored graph on n vertices and E(G) 6= ;. If dc_{(u) +} dc(v) ≥ n + 1 for every edge uv ∈ E(G), then G contains a rainbow triangle.

In fact, the color degree sum bound “n+ 1” is sharp for the existence of rainbow triangles. This can be shown by the following two classes of colored graphs.

Example 2.2. Let K_k_,n_−k(1_{≤ k ≤ n/2) be a PC complete bipartite graph.} Example 2.3. Let D_nbe a colored graph with V(D_n) = {u₁, u₂, v₁, v₂, . . . , v_n₋₂_}, E(Dn) = {u1u2} ∪ {uivj : i = 1, 2; j = 1, 2, . . . , n − 2}, and col(u1u2) =

0, col(uivj) = j, (i = 1, 2; j = 1, 2, . . . , n − 2).

It is easy to check that both examples satisfy dc_{(u) + d}c_{(v) ≥ n for every}

edge uv but contain no rainbow triangles. Let_Gc

1 be the set of all PC complete bipartite graphs andG2cbe the set of

all D_n-type graphs. With more efforts, we can prove thatGc

1 andG2c are the

only classes of extremal graphs when lowering the bound “n+ 1” to “n”. Theorem 2.5. Let G be a colored graph on n _{≥ 5 vertices and E(G) 6= ;.} If dc(u) + dc(v) ≥ n for every edge uv ∈ E(G) and G contains no rainbow triangles, then G∈ Gc

1∪ G2c.

Here the condition E(G) 6= ; in above theorems is necessary. If E(G) is empty, then the restrictions on the color degree sum of adjacent vertices are meaningless.

2.2 Two lemmas

Before presenting the proofs of the main results, we first prove the following lemmas.

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Lemma 2.6. Let G be a colored graph on n vertices and E(G) 6= ;. If G is triangle-free and dc(u) + dc(v) ≥ n for every edge uv ∈ E(G), then G is a complete bipartite graph with a proper edge-coloring.

Proof. Since G contains no triangles, for each edge uv ∈ E(G), we have N(u) ∩ N(v) = ;. So d(u) + d(v) ≤ n. Also, d(u) + d(v) ≥ dc(u) + dc(v) ≥ n. Hence d(u) + d(v) = dc_{(u) + d}c_{(v) = n. This implies that G is properly}

colored.

Let x y be an edge in G and N(x) = A. Then N(y) = V (G)\A. Let N(y) = B. Then y ∈ A and x ∈ B. Since G is triangle-free, G[A] and G[B] are empty graphs. For each vertex a_{∈ A, we have ax ∈ E(G) and N (a) ⊆ B.} Thus,

|B| ≥ d(a) ≥ dc(a) = n − dc(x) = n − d(x) = n − |A| = |B|.

This implies that N(a) = B. Similarly, for each vertex b ∈ B, we have N(b) = A.

Hence G = (A, B) is a complete bipartite graph with a proper edge-coloring.

Lemma 2.7. Let G be a colored graph on n ≥ 6 vertices such that dc_{(u) +} dc(v) ≥ n for every edge uv ∈ E(G). Let x be a vertex in G such that dc(x) = δc_{(G) and let G}0_{= G − x. If G}0 _{is a PC complete bipartite graph and G is not}

triangle-free, then G contains a rainbow triangle.

Proof. Let G0be a PC K_k_,n_−1−k= (A, B). Then for vertices a ∈ A and b ∈ B, we have d_Gc₀(a) = n− k −1 and d_Gc₀(b) = k. Let A0= N(x)∩A and B0= N(x)∩ B. Since G is not triangle-free, we have A0, B0_{6= ;.}

Claim 1. For vertices a _{∈ A}0 and b _{∈ B}0, dc

G0(a) ≥ n/2 − 1 and d c G0(b) ≥ n/2 − 1.

Proof. Since dc(a) ≥ dc(x) ≥ n − dc(a) and dc(b) ≥ dc(x) ≥ n − dc(b), we have dc(a) ≥ n/2 and dc(b) ≥ n/2. So we obtain d_Gc₀(a) ≥ dc(a)−1 ≥ n/2−1 and dc

G0(b) ≥ dc(b) − 1 ≥ n/2 − 1.

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2.2. Two lemmas 17

Proof. Choose a_{∈ A}0and b_{∈ B}0. Then

d_Gc0(a) + d_Gc0(b) = n − 1. (2.2)

If n is odd, then n_{≥ 7. By Claim 1 and Equation (2.2), d}_Gc₀(a) = d_Gc₀(b) = (n − 1)/2. Thus dc_{(b) ≤ d}c

G0(b) + 1 = (n + 1)/2. So dc(x) ≥ n − dc(b) ≥

(n − 1)/2 ≥ 3.

If n is even, then by Claim 1 and Equation (2.2), we have min_{d_Gc₀(a), d_Gc₀(b)} = n/2 − 1.

Thus, min{dc_{(a), d}c_{(b)} ≤ min{d}c G0(a), d

c

G0(b)} + 1 = n/2. So d

c_{(x) ≥ n −}

min_{dc(a), dc(b)} ≥ n/2 ≥ 3.

Claim 2 implies that there exist a₁_{∈ A}0and b₁_{∈ B}0such that col(xa₁) 6= col(x b1). Let col(xa1) = 1 and col(x b1) = 2. Now, we will prove this lemma

by contradiction.

Suppose that G contains no rainbow triangles. Then col(a₁b₁) ∈ {1, 2}. Without loss of generality, assume that col(a₁b1) = 1. Then dc(a1) = d_Gc0(a1).

Hence, for each vertex b∈ B, we have dc_{(b) ≥ n − d}c_(a

1) = n − d_Gc0(a1) =

d_Gc0(b) + 1. Thus B0= B and dGcol(x b)(b) = 1.

Since_|B0_{| = |B| = d}c

G0(a1) ≥ n/2 − 1 ≥ 2, we have B

0_\{b

1} 6= ;. Let b be

a vertex in B0_\{b₁_{}. Consider the triangle x a}₁b x. Since d_Gcol(x b)(b) = 1 and

G0 is properly colored, we have col(x b) = col(xa₁) = 1. This means that col(x b) = 1 for every vertex b ∈ B0\{b1}.

Furthermore, by Claim 2, there is a vertex a₂_{∈ A}0such that col(xa₂) 6∈ {1, 2}. Let col(x a2) = 3. Let b2 be a vertex in B0\{b1}. Then col(x b2) = 1.

Since the triangle x a2b1x is not rainbow and dGcol_{(x b1)}(b₁) = 1, we have col(a2b1) = 3. Similarly, considering the triangle xa2b2x and the fact that

d_Gcol_{(x b2)}(b₂) = 1, we get col(a₂b₂) = 3. This contradicts that G0 is a PC

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2.3 Proofs of theorems

Proof of Theorem 2.4. Suppose the contrary. Assume that G is a counterex-ample such _{|V (G)| + |E(G)| as small as possible. Let x y be an edge of G.} Then

n− 1 ≥ max{dc(x), dc(y)} ≥ (dc(x) + dc(y))/2 ≥ (n + 1)/2.

This implies that n _{≥ 3. If δ}c_{(G) ≥ (n + 1)/2, then by Theorem 2.1, G}

contains a rainbow triangle, a contradiction. So there must be a vertex x ∈ V(G) such that dc(x) < (n + 1)/2. Let G0= G − x.

Claim 1. E(G0) is nonempty.

Proof. If d(x) = 0, then there is nothing to prove. If d(x) > 0, then there exists a vertex y_{∈ N (x) such that d( y) ≥ d}c(y) ≥ n+1−dc(x) > (n+1)/2 ≥ 2. So d_G0(y) = d(y) − 1 > 1. This shows that E(G0) is nonempty.

Claim 2. For each edge uv_{∈ E(G}0), dc

G0(u) + d_Gc0(v) ≥ n.

Proof. If u_{6∈ N (x) or v 6∈ N (x), then d}_Gc₀(u) + d_Gc₀(v) ≥ dc(u) + dc(v) − 1 ≥ n. If u, v ∈ N (x), then dc_{(u) > (n + 1)/2 and d}c_{(v) > (n + 1)/2. Thus} d_Gc₀(u) + d_Gc₀(v) ≥ dc(u) + dc(v) − 2 > n − 1. Hence, d_Gc₀(u) + d_Gc₀(v) ≥ n.

By Claims 1 and 2, G0is a smaller counterexample, a contradiction. This completes the proof.

Proof of Theorem 2.5. Case 1. n= 5.

If G is triangle-free, then by Lemma 2.6, G is a PC complete bipartite graph, thus G _{∈ G}₁c. Now, suppose that G contains a triangle. Let S= {v : dc(v) ≤ 2} and T = {v : dc(v) ≥ 3}.

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2.3. Proofs of theorems 19

Proof. Since dc_(u)+dc_{(v) ≥ 5 for every edge uv ∈ E(G), S is an independent}

set. Furthermore, we have|T | ≥ 1 by the fact that E(G) 6= ;. If |T | = 1, then Gis a bipartite graph. This contradicts that G contains a triangle. So we have |T | ≥ 2. Now we will prove that T is a clique by contradiction.

Suppose that there are vertices u, v_{∈ T such that uv 6∈ E(G). Then d(u) =} d(v) = 3 and dc(u) = dc(v) = 3. Let {x, y, z} = V (G)\{u, v}, col(ux) = 1, col(uy) = 2 and col(uz) = 3. Since G is not a bipartite graph, the edge set of G[{x, y, z}] is nonempty. So there exists a vertex in {x, y, z}, say x, satisfying that dc_{(x) ≥ 3. Furthermore, there is a vertex s ∈ {y, z} such that} xs∈ E(G) and col(xs) 6= 1. Without loss of generality, assume that s = y. Then col(x y) = 2. Now consider the triangle vx y v. We have col(x v) = 2 or col(y v) = 2.

If col(x v) = 2, then xz ∈ E(G) and col(xz) = 3. Now, xzvx is a triangle and col(zv) 6= col(x v). So col(vz) = col(xz) = 3. Note that dc(z) ≥ 5 − dc(v) = 2. So yz ∈ E(G) and col(yz) 6= 3. Since x yzx is a triangle but not rainbow, we have col(yz) = 2. Thus, dc(y) ≤ 2 and dc(y) + dc(z) ≤ 4 < 5 for the edge yz, a contradiction.

If col(y v) = 2, then dc(y) ≤ 2. Furthermore, we have dc(y) ≥ 5 − dc(u) = 2. So dc(y) = 2. This implies that yz ∈ E(G) and col(yz) = 3. Since dc(z) ≥ 5 − dc(y) = 3, we have col(vz) 6= 3. Consider the triangle yzv y. We have col(zv) = 2. However, this contradicts that col(v y) 6= col(vz).

In summary,_{|T | is a clique.} Claim 2. _{|T | = 2.}

Proof. By contradiction.

If _{|T | = 5, by Theorem 2.1, G contains a rainbow triangle, a} contradic-tion.

If _{|T | = 4, by Claim 1, G[T ] ∼}_{= K}₄. We first prove that d_Gc_[T](v) = 2 for every vertex v _{∈ T . Since 3 ≥ d}c

G[T](v) ≥ dc(v) − 1 ≥ 2, it is sufficient to

show that d_Gc_[T](v) 6= 3 for every vertex v ∈ T. Suppose that this is false. Then there is a vertex v₀_{∈ T such that d}_Gc_[T](v₀) = 3. Let T = {v₀, v₁, v₂, v₃_}. Without loss of generality, assume that col(v0vi) = i (i = 1, 2, 3) and let col(v₁v₂) = 1. Since d_Gc_[T](v_i) ≥ 2 (i = 1, 3), we have col(v₁v₃) = 3

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and col(v₃v2) = 2 by considering triangles v0v1v3v0 and v0v2v3v0. Thus,

we obtain a rainbow triangle v1v2v3v1, a contradiction. So for every

ver-tex v _{∈ T , d}_Gc_[T](v) = 2. Let {x} = V (G)\T. We have col(x v_i) ∈ E(G) and d_Gcol_{(x vi)}(v_i) = 1 (i = 0, 1, 2, 3). Since G contains no rainbow

trian-gles, we have col(x v_i) = col(x v_j) (i, j = 0, 1, 2, 3). Thus dc(x) = 1 and dc(x) + dc(v0) = 4 < 5, a contradiction.

If |T | = 3, then set T = {x, y, z} and S = {u, v}. By Claim 1, x yz x is a triangle and uv _{6∈ E(G). Without loss of generality, assume that col(x y) =} col(xz) = 1, col(ux) = 2 and col(vx) = 3. We have dc(x) = 3 and dc(u) = dc(v) = 2. Thus, there exists a vertex s ∈ {y, z} such that col(us) 6= col(ux). Combining this with the fact that col(ux) 6= col(x y) and col(ux) 6= col(xz), we have col(us) = col(xs). Without loss of generality, assume that s = y. Then col(uy) = 1. Now, consider that dc(y) ≥ 3 and dc(v) = 2. We have col(y v) = col(x v) = 3 and col(vz) = col(xz) = 1. Note that the edge yz is contained in the triangle v yz v. So col(yz) = 1 or 3. However, this implies that dc(y) ≤ 2, a contradiction.

Thus, we have_{|T | ≤ 2. By Claim 1, we get |T | = 2.}

Now, let T = {u, v} and S = {x, y, z}. By Claim 1, uv ∈ E(G) and S is an independent set. If dc(x) = dc(y) = dc(z) = 1, then dc(u) = dc(v) = 4. Thus, obviously, G _{∈ G}₂c. If there is a vertex in S, say x, satisfying dc(x) = 2, then col(xu) 6= col(x v). Since xuvx is not a rainbow triangle, we can assume that col(xu) = col(uv). Thus we have yu, zu ∈ E(G), dc(u) = 3, col(yu) 6= col(uv), col(zu) 6= col(uv) and dc(y) = dc(z) = 2. Since yuv y and zuvz are not rainbow triangles, we have col(y v) = col(zv) = col(uv). This implies that dc(v) ≤ 2, a contradiction.

Case 2. n≥ 6.

We prove this case by induction. Note that Theorem 2.5 is true for graphs on 5 vertices. Assume that it is true for graphs of order n_{− 1 (n ≥ 6). We} will prove that it is also true for graphs of order n.

Let G be a graph on n≥ 6 vertices. Since G contains no rainbow triangles, by Theorem 2.1, we haveδc(G) ≤ n/2. If δc(G) = n/2, by Theorem 2.3, n is even and G is a PC K_n_/2,n/2. If G is triangle-free, by Lemma 2.6, G is a

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2.3. Proofs of theorems 21

complete bipartite graph with a proper edge-coloring. In both cases, we have G∈ Gc

1.

Now, consider the case thatδc(G) < n/2 and G is not triangle-free. Let x be a vertex in G such that dc_{(x) = δ}c_{(G). Let G}0_{= G− x. Similar to the proof}

of Theorem 2.4, we have E(G0) 6= ; and d_Gc₀(u) + d_Gc₀(v) ≥ n − 1 for every edge uv _{∈ E(G}0). This implies that G0 satisfies the conditions in Theorem 2.5. By assumption, G0∈ Gc

1 ∪ G2c. However, by Lemma 2.7, G0is not a PC

bipartite graph. Hence, G0_{∈ G}₂c. Now, we will prove that G _{∈ G}₂c. Without loss of generality, let

V(G0) = {u1, u2, v1, v2, . . . , vn−3}, E(G0) = {u₁u₂_{} ∪ {u}_iv_j: i= 1, 2; j = 1, 2, . . . , n − 3}, and col(u₁u₂) = 0, col(u_iv_j) = j (i = 1, 2; j = 1, 2, . . . , n − 3). Thus, we have d_Gc0(u1) = d_Gc0(u2) = n − 2 and d_Gc0(vi) = 1 (i = 1, 2, . . . , n − 3). Since dc(x) + dc(v_i) ≤ 2dc(v_i) ≤ 2d_Gc0(vi) + 2 = 4 < n (i = 1, 2, . . . , n − 3), we have N(x) ⊆ {u1, u2} and dc(v_i) = d_Gc0(vi) = 1 (i = 1, 2, . . . , n − 3). Furthermore, we get n_{≤ d}c(u_j) + dc(v₁) ≤ (d_Gc0(uj) + 1) + 1 = n (j = 1, 2).

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This implies that dc(u_j) = d_Gc0(uj) + 1 (j = 1, 2). Thus, {u1, u2} ⊆ N (x) and 1≤ dc(x) ≤ dc(v1) = 1.

Now, N(x) = {u₁, u₂_{}, d}c(u₁) = dc(u₂) = n − 1 and dc(x) = dc(v_i) = 1 for i= 1, 2, . . . , n − 3. This implies that G ∈ G₂c.

The proof is complete.

2.4 Rainbow triangles in edge-colored complete graphs

Gallai gave a characterization of edge-colored complete graphs without con-taining rainbow triangles.

Theorem 2.8 (Gallai[32]). For an edge-colored complete graph K_n, if it does not contain a rainbow triangle, then V(K_n) can be partitioned into several (at least2) parts such that between the parts, there are a total of at most two colors and, between every pair of parts, there is only one color on the edges.

Based on this structure, Gyárfás and Simonyi[34] proved that each edge-colored complete graph K_nwith∆mon_(K

n) < 2n₅ contains a rainbow triangle,

and this bound is tight. In this section, a color degree condition for the existence of rainbow triangles in edge-colored complete graphs is obtained by two different proofs. One is based on Theorem 2.8, the other is self-contained.

Theorem 2.9. Ifδc(K_n) > log₂n with n≥ 3, then this colored Kn contains a rainbow triangle.

Remark 2.1. The bound of δc(K_n) in Theorem 2.9 is tight. The following construction due to Li and Wang [41] shows the sharpness. Let G₁ = K₂. For the unique edge e_{∈ E(G}₁), let col(e) = 1. For i = 2, 3, . . ., construct an edge-colored graph G_i₊₁by joining two disjoint copies of G_i completely with

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2.4. Rainbow triangles in edge-colored complete graphs 23

edges of color i+ 1. The resulting edge-colored complete graph K_n satisfies δc_(K

n) = log2nbut it has no rainbow triangle.

Proof of Theorem 2.9 by applying Theorem 2.8. We prove the theorem by induction on n. It is easy to check that the theorem holds for small n. So we assume that n_{≥ 5.}

Suppose that K_n contains no rainbow triangle. Then it follows from The-orem 2.8 that K_n has a Gallai partition S₁, S₂, . . . , S_k with k _{≥ 2. If k ≥ 4} then there exists S_j such that_|S_j_{| ≤} n

4. Sinceδ

c_(K

n[Sj]) > log2|Sj|, applying

induction hypothesis to K_n[S_j], we can find a rainbow triangle, a contradic-tion. Thus we may assume that 2_{≤ k ≤ 3. In this case, we can take S}_jso that only one color is used on edges between S_j and V(K_n) \ S_j. Again, applying induction hypothesis to the smaller part of _{S_j, V(K_n) \ S_j_{}, we can find a} rainbow triangle, a contradiction. Hence the theorem holds.

Proof of Theorem 2.9 without applying Theorem 2.8. Let K be a colored K_n. If n= 3 or 4, by simple calculation, K is a PC graph. So it is trivial that K contains a rainbow triangle. For n_{≥ 5, we prove by contradiction. Suppose} that K is a counterexample to Theorem 2.9 on smallest number of vertices. Let M = min{k ∈ N : k > log₂n}. Then dc_{(u) ≥ M for each vertex u ∈ V (K).}

Let u be a vertex in K and H = K − u. Then there must exist a vertex v ∈ V (H) such that dc

H(v) ≤ log2(n − 1). Otherwise, there is a rainbow

triangle in H_{⊂ K.}

Since d_Hc(v) ≥ dc(v) − 1, we have

M− 1 ≤ dc(v) − 1 ≤ dHc(v) ≤ log2(n − 1) < log2n< M.

This implies that M−1 = dc_{(v)−1 = d}c

H(v). Thus col(uv) appears only once

at v and dc(v) = M. Here, for convenience, we say a vertex u dominates a vertex v (or u₋_{→ v) if col(uv) appears only once at v and d}D c_{(v) = M. Now}

we have Claim 1.

Claim 1. For every vertex u_{∈ V (K), there is at least one vertex v ∈ V (K)} such that u−→ v.D

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Claim 2. For every vertex v _{∈ V (K), there is at most one vertex u ∈ V (K)} such that u−→ v.D

Proof. Suppose the contrary. Then there exist vertices v, u1and u2 (u16= u2)

satisfying u₁₋_{→ v and u}D ₂₋_{→ v. Assume that col(u}D ₁v) = α and col(u2v) = β.

Thenα 6= β. Since u₁u₂vu₁ is not a rainbow triangle, we have col(u₁u₂) ∈ {α, β}. Without loss of generality, let col(u1u2) = α. Now, for any vertex

u ∈ V (K)\{u1, u2, v}, col(vu) 6∈ {α, β} and col(u1u) ∈ {col(vu), α}. This

implies that Nc(u₁) ⊆ Nc(v)\{β}. Hence we have

dc(u1) = |Nc(u1)| ≤ |Nc(v)| − 1 = M − 1,

a contradiction.

Now, construct a directed graph H with V(H) = V (K) satisfying that A(H) = {uv : u−→ v}.D

Claims 1 and 2 show that d_H+(u) ≥ 1 and d_H−(u) ≤ 1 for each u ∈ V (H). This implies that d_H+(u) = d_H−(u) = 1 for each u ∈ V (H). Thus, every component H_i of H is a directed cycle and also a PC cycle in K unless|V (Hi)| = 2. Let Hi = v1v2· · · vkv1. Since there is no rainbow triangle in K, we have k= 2 or

k≥ 4.

Assume that H_i is a component of H with _{|V (H}_i)| ≥ 4, i.e., k ≥ 4, we have the following claims.

Claim 3. For any vertices v_s, v_t∈ V (Hi), col(vsvt) ∈ col(Hi).

Proof. Suppose the contrary. Let v_s, v_t_{∈ V (H}_i) be a pair of vertices satisfying col(v_sv_t) 6∈ col(H_i) and |t − s| = min{|t0− s0| : col(vs0vt0) 6∈ col(Hi), vs0, vt0∈ V(Hi)}. Without loss of generality, assume that s < t.

If s= t − 1, then v_sv_t∈ A(Hi), thus col(vsvt) ∈ col(Hi). If s = t − 2, since v_sv_s₊₁v_tv_s is not a rainbow triangle, there holds

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So we have s_{≤ t − 3. Consider the triangle v}_svs+1vtvs. Since vsvs+1∈ A(Hi), col(v_sv_s₊₁) appears only once at v_s₊₁ and col(v_sv_t) 6∈ col(H_i), we have col(vsvs+1) 6= col(vs+1vt) and col(vsvt) 6= col(vsvs+1). This implies that col(v_sv_t) = col(v_s₊₁v_t). Thus col(v_s₊₁v_t) 6∈ col(H_i) and 2 ≤ t−(s+1) < t−s, a contradiction.

Claim 4. For each colorα ∈ col(H_i), there are at least two arcs e, f ∈ A(H_i) such that col(e) = col(f ) = α.

Proof. Suppose the contrary. There is a color α ∈ col(H_i) such that only one arc, say v₁v2 satisfying col(v1v2) = α. Since col(v1v2) 6= col(v2v3),

col(v2v3) 6= col(v1v3) and there is no rainbow triangle in K, we must have

col(v₁v₃) = col(v₁v₂) = α. Similarly, for j = 4, . . . , k − 1, consider the color-ing of the triangle v₁vj−1vjv1, we can get that col(v1vj) = col(v1vj−1) = α.

However, this produces a rainbow triangle v1vk−1vkv1, a contradiction.

Claim 5. For each colorα ∈ col(H_i), if col(v_sv_s₊₁) = α, then col(v_s₊₂v_s₊₃) = α (indices are taken modulo k).

Proof. Suppose the contrary. By Claim 4, there are two arcs v_sv_s₊₁, v_tv_t₊₁and a colorα ∈ col(H_i) satisfying col(v_svs+1) = col(vtvt+1) = α with s ≤ t − 3

and for any r∈ [s +1, t −1], col(vrvr+1) 6= α. Thus the color col(vsvs+1) = α

appears only once on the directed path v_sv_s₊₁_{· · · v}_t. By a similar argument as that in the proof of Claim 4, we have col(v_svs+1) = col(vsvs+2) = · · · =

col(v_sv_t₋₁) = col(v_sv_t) = α. Note that s ≤ t − 3. So t − 2 ≥ s + 1. Let col(v_t₋₂v_t₋₁) = β and col(v_t₋₁v_t) = γ. Then α, β and γ are three dis-tinct colors. It is easy to see that col(v_t₋₂vt+1) = β and col(vt−1vt+1) = γ.

Now, consider triangles v_sv_t₋₂v_t₊₁v_sand v_sv_t₋₁v_t₊₁v_s, we have col(v_sv_t₊₁) ∈ {α, β} ∩ {α, γ} = {α}. However this contradicts that col(vtvt+1) = α appears

only once at v_t₊₁.

Claim 5 implies that _|col(H_i)| = 2 and |V (H_i)| is even. Let col(H_i) = {α, β}. Then these two colors appear alternatively on Hi.

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Proof. Suppose that_{|V (H}_i)| ≥ 6. Let col(v₁v2) = α. Then col(v4v5) = β.

Since v1v2, v4v5 ∈ A(Hi), we have col(v2v5) 6= col(v1v2) and col(v2v5) 6=

col(v4v5). Thus col(v2v5) 6∈ {α, β} = col(Hi). This contradicts Claim 3.

Now, it is clear that every component of H is either a dicycle of length 2 or a dicycle of length 4 assigned two colors. Let H₁, H₂, . . . , H_s_+t be the components of H. Assume that|V (Hi)| = 2 for i ∈ [1, s] and |V (Hj)| = 4 for j∈ [s + 1, s + t]. Let Hi= ui1ui2ui1and Hj= vj1vj2vj3vj4vj1for i= 1, 2, . . . , s

and j= s + 1, s + 2, . . . , s + t. Let G be a subgraph of K with

V(G) = {ui1: i= 1, 2, . . . , s} ∪ {vj1, vj2: j= s + 1, s + 2, . . . , s + t},

E(G) = {uv : u, v ∈ V (G)}. Claim 7. For every vertex u_{∈ V (G), d}_Gc(u) > log₂_{|V (G)|.}

Proof. We will prove that for all a∈ V (Hi) ∩ V (G) and all b ∈ V (Hj)\V (G), i, j = 1, 2, . . . , s + t, i 6= j, there is a vertex w ∈ V (G) such that col(aw) = col(ab).

Case 1. b= u_j₂for some j_{∈ [1, s].}

Let col(u_j₁uj2) = α. Since the color α appears only once at uj1 and also

at u_j2. We have col(auj1) 6= α and col(auj2) 6= α. Consider the triangle

au_j₁u_j₂a, we get col(au_j₁) = col(ab). Choose w = u_j₁. Case 2. b= v_j₃ or v_j₄ for some j_{∈ [s + 1, s + t].}

Assume that b= v_j3. If col(ab) 6∈ col(Hj), we have col(avj4) = col(ab)

and col(av_j₁) = col(ab). In this case, choose w = v_j₁. If col(ab) ∈ col(H_j), by Claim 5, we can assume that col(v_j₁vj2) = col(vj3vj4) = α and col(vj4vj1) =

col(v_j₂v_j₃) = β. Since col(ab) 6= col(v_j₂v_j₃) and col(av_j₂) 6= col(v_j₁v_j₂), we have col(ab) = α and col(av_j₂) 6= α. Now, consider triangles av_j₂vj3a and

av_j2vj1a, we can obtain that col(avj2) = β and col(avj1) = α = col(ab). In

this case, let w = v_j₁. In conclusion, when b = v_j₃, we can always choose w= vj1. When b= vj4, by a similar argument, we can choose w= vj2.

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Now, for all i= 1, 2, . . . , s and j = s + 1, s + 2, . . . , s + t, it is clear that N_Gc(u_i1) = Nc(ui1)\{col(ui1ui2)},

N_Gc(v_j₁) = Nc(v_j₁)\{col(v_j₄v_j₁)}, and

N_Gc(v_j₂) ⊇ Nc(v_j₁)\{col(v_j₂v_j₃)}. So we have dc

G(u) ≥ dc(u) − 1 > log2n− 1 = log2(n/2) = log2|V (G)| for

every vertex u∈ V (G).

Note that K is a minimum counterexample to Theorem 2.9. Claim 7 implies that there is a rainbow triangle in G and which is also a rainbow triangle in K, a contradiction.

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Chapter 3

PC C

₄

’s in edge-colored

complete bipartite graphs

3.1 Introduction

Throughout this chapter, we use C_`to denote a cycle of length`.

When considering the existence of PC cycles in an edge-colored graph, one often needs to know the structure of graphs containing no PC cycles. So we start with the following important structural result:

Theorem 3.1 (Grossman and Häggkvist[33], Yeo [65]). Let G be an edge-colored graph containing no PC cycles. Then there is a vertex z ∈ V (G) such that no component of G− z is joint to z with edges of more than one color.

There are lots of results and problems on the existence of PC Hamilton cycles and long cycles (See[5,20,28,45,46,53,62]). For short PC cycles, es-pecially, a PC triangle (or a rainbow triangle), the well-known Gallai coloring theory gives a structural characterization of edge-colored complete graphs containing no rainbow triangles (See [32] and [34]). Conditions for the existence of rainbow triangles in edge-colored graphs (not necessarily com-plete) are given in[38] and [39]. In a variety of those work, minimum color degree conditions for edge-colored graphs to contain PC cycles with certain properties are often discussed. One natural problem in this area would be

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asking a sharp color degree condition for an edge-colored graph to contain a PC cycle. In this paper, we first give a complete solution to this problem.

To state our answer, we construct an edge-colored graph G_D such that δc_(G

D) = D and GDcontains no PC cycles. Let G1 be an edge-colored graph

which is isomorphic to K₂ with color cG1

1 . For 1 ≤ i ≤ D − 1, let Gi+1 be

an edge-colored graph obtained from i+ 1 vertex-disjoint copies of G_i, say H1, . . . , Hi+1, and a new vertex vi+1by joining vi+1to each Hjfor 1≤ j ≤ i+1

and coloring the edges from v_i₊₁ to H_j with color cHj

i+1 (for s 6= t, col(Hs)

and col(H_t) may be different as long as G_i₊₁ satisfies the condition that δc_(G

i+1) = i + 1 and Gi+1 has no PC cycles). By the construction, we can

easily check that|V (GD)| = D! P D i=0 1i!,δ

c_(G

D) = D and GDhas no PC cycles.

Theorem 3.2. Let G be an edge-colored graph of order n with δc_{(G) ≥ D.} Suppose that n_{≤ D!}PD_i₌₀_i1_! and G contains no PC cycles. Then the equality on n is attained, and moreover, G is isomorphic to GD, up to the edge-coloring structure.

The problem of giving sharp minimum color degree conditions for short PC cycles seems more difficult. So far, we have some known results which give partial answers to this problem. Lo[45] showed that, for any constant number ε > 0, if an edge-colored graph of order n (sufficiently large) has minimum color degree at least(2₃+ ε)n, then it contains a PC cycle of length ` for all 3 ≤ ` ≤ n. Li [39] proved that if an edge-colored graph of order n has minimum color degree at least n+1₂ , then it contains a PC triangle. ˇ

Cada et al.[19] proved that if an edge-colored triangle-free graph of order nhas minimum color degree at least n

3+ 1 then it contains a rainbow C4. As

seen in these results, it seems a reasonable approach for us to consider this problem for some specified graph classes. In this paper we shall restrict our considerations to this problem in edge-colored complete graphs and complete bipartite graphs.

In the study of PC Hamilton cycles and long cycles in edge-colored com-plete graphs, maximum monochromatic degree conditions are often involved (See[47] and [62] and the papers cited therein). Bollobás and Erd˝os [15] conjectured that every edge-colored complete graph K_nwith∆mon_(K

n) < bn₂c

Properly colored cycles in edge-colored graphs

P

ROPERLY

C

OLORED

C

YCLES IN

E

DGE

-C

OLORED

G

RAPHS

PROPERLY COLORED CYCLES IN

EDGE-COLORED GRAPHS

http://doi.org/10.3990/1.9789036544719

Preface

Papers underlying this thesis

Contents

Chapter 1

Introduction

1.1

Terminology and notation

1.2

PC tours and PC cycles

1.2.1

Short PC cycles

1.2.2

Long PC cycles

1.3

PC cycles and directed cycles

Chapter 2

Rainbow triangles in

edge-colored graphs

2.1

Introduction

2.2

Two lemmas

2.3

Proofs of theorems

2.4

Rainbow triangles in edge-colored complete graphs

Chapter 3

PC C

4

’s in edge-colored

complete bipartite graphs

3.1

Introduction

₄