© 2019 The Authors. Journal of Graph Theory published by Wiley Periodicals, Inc. J Graph Theory. 2020;94:476–493. 476
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DOI: 10.1002/jgt.22536 A R T I C L EVertex
‐disjoint properly edge‐colored cycles
in edge
‐colored complete graphs
Ruonan Li
1,2|
Hajo Broersma
3|
Shenggui Zhang
1,21Department of Applied Mathematics,
Northwestern Polytechnical University, Xi’an, China
2
Xi’an‐Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, China
3
Faculty of EEMCS, University of Twente, Enschede, The Netherlands Correspondence
Hajo Broersma, Faculty of EEMCS, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Email: h.j.broersma@utwente.nl Funding information China Scholarship Council,
Grant/Award Number: 201506290097; National Natural Science Foundation of China, Grant/Award Number: 11271300; CSC, Grant/Award Number:
201506290097; NSFC,
Grant/Award Numbers: 11671320, 11901459; The Fundamental Research Funds for the Central Universities of China, Grant/Award Numbers: 31020180QD124, 3102019GHJD003
Abstract
It is conjectured that every edge‐colored complete graph G on n vertices satisfyingΔmon( )G ≤ n− 3 + 1k containsk vertex‐disjoint properly edge‐colored cycles. We confirm this conjecture fork = 2, prove several additional weaker results for generalk, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge‐colored cycles in edge‐colored complete graphs and directed cycles in multipartite tournaments. Using this relationship and our results on edge‐colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.
K E Y W O R D S
complete graph, edge‐colored graph, multipartite tournament, properly edge‐colored cycle, vertex‐disjoint cycles
J E L C L A S S I F I C A T I O N 05C15; 05C20; 05C38
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I N T R O D U C T I O N
All graphs considered in this paper are finite and simple. For terminology and notation not defined here, we refer the reader to [6].
Let G be a graph with vertex setV G( ) and edge set E G( ). For a nonempty subset S of V G( ), let G S[ ] denote the subgraph of G induced by S, and let G−S denote the subgraph of G induced byV G( )⧹S. When S= { }, we use Gv −vinstead of G− { }. An edgev ‐coloring of G is a -This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
mapping col E G: ( ) →, where is the set of natural numbers. A graph G with an edge‐ coloring is called an edge‐colored graph or simply a colored graph. We say that a colored graph G is a properly colored graph or simply a PC graph if each pair of adjacent edges (ie, edges incident with one common vertex) in G are assigned distinct colors. A PC graph G is called a rainbow graph if all the edges of G are assigned different colors.
Let G be a colored graph. For a vertexv ∈V G( ), the color degree of v, denoted by d vGc( )is the number of different colors appearing on the edges incident with v. For an edgee ∈E G( ), let colG( )e denote the color of e. For a subgraph H of G, let colG( )H denote the set of colors appearing onE H( ). For two vertex‐disjoint subgraphs F andH of G, letcolG( ,F H)denote the set of colors appearing on the edges between F and H. If V F( ) = { }, then we often writev colG( ,v H)instead ofcolG( ,F H). For two disjoint nonempty subsets S andT ofV G( ), we use colG( , )S T as shorthand for colG( [ ],G S G T[ ]). When there is no ambiguity, we often writed vc( ) for dGc( )v , col e( ) for colG( )e , col H( ) for colG( )H , col F H( , ) for colG( ,F H), col v H( , ) for colG( ,v H), and col S T( , ) for colG( , )S T . For each colori∈ col G( ), we use Gi to denote the spanning subgraph of G induced by the edges of coloriin G. LetΔmon( )G denote the maximum monochromatic degreeof G, that is,Δmon( ) = max{Δ( ):G Gi i ∈col G( )}. Throughout this paper, we use C3 and C4 to denote cycles of length 3 and 4, respectively. We also frequently use the term triangle for a C3.
Research problems related to PC cycles and rainbow cycles in colored graphs have attracted a lot of attention during the past decades, not only because of the many challenging open problems and conjectures and interesting results, but also because of the relation to problems on cycles in digraphs. Here we adopt the terminology of [4] and use digraph for directed graph and cycle for a directed cycle in a digraph. We refer the reader to [10] and Chapter 16 in [4] for surveys on rainbow cycles and PC cycles, respectively. We also recommend proof techniques in [11,12], constructions in [7], and Chapter 16 in [4] for a glance of the deep relation between edge‐colored graphs and digraphs. Here, we are mainly interested in the existence of vertex‐ disjoint PC cycles (called disjoint PC cycles for simplicity in the sequel) in colored complete graphs. Our first easy observation implies that having k disjoint PC cycles is equivalent to havingk disjoint PC cycles of length at most 4 in colored complete graphs.
Observation1. Let G be a colored Knand letCbe a PC cycle in G. Then, for each vertex ∈
v V C( ), there exists a PC cycle C′ of length at most 4 in G containing v and with ⊆
V C( ′) V C( ).
Proof. Let G′ =G V C[ ( )]. It is equivalent to prove that each vertex in G′ is contained in a PC cycle of length at most 4.
By contradiction. Suppose that C′ =v v1 2…v vr 1is a PC cycle of minimal length inG′ containing v1, and assume thatr≥5. Assume without loss of generality that col v v( 1 2) = 1 and col v v( 1 r) = 2. If col v v( 3 4) = 1, then, using the chord v v1 4, either v v1 2…v v4 1 or v v v1 r r−1…v v4 1is a PC cycle inG′ containing v1 and shorter thanC′, a contradiction. So,
≠
col v v( 3 4) 1. Similarly, using the chord v v1 3, we can show that col v v( 3 4)≠2. Without loss of generality, assume that col v v( 3 4) = 3. Ifcol v v( 1 3) = 3, then v v v v1 2 3 1is a PC cycle in G′ containing v1and shorter thanC′, a contradiction. So,col v v( 1 3)≠3. Similarly, we can show thatcol v v( 1 4)≠ 3by considering the cycle v v v1 r r−1…v v4 1. Noting thatC′ is a shortest PC cycle containing v1inG′, the cycles v v v v v1 2 3 4 1and v v v1 r r−1 4 3 1v v v are not PC cycles. So we have col v v( 1 4) = 1and col v v( 1 3) = 2. This implies that v v v v1 3 4 1is a PC C3, our final
Before turning to disjoint PC cycles, we first recall the following fundamental result on the existence of PC cycles in colored graphs.
Theorem 2 (Grossman and Häggkvist [9] and Yeo [13]). Let G be a colored graph containing no PC cycles. ThenG contains a vertex v such that no component of G−v is joined to v with edges of more than one color.
Combining Theorem 2 and Observation 1 for colored complete graphs, we immediately obtain a maximum monochromatic degree condition for the existence of a PC C3 or C4.
Observation3. Let G be a colored Kn. IfΔmon( )G ≤n− 2, then G contains a PC cycle of length at most 4.
The observation follows from the simple fact that in a complete graph G on at least two vertices, for every vertex v of G, G−v consists of only one component.
Using Observations 1 and 3, and repeatedly deleting the vertices of PC cycles of length at most 4, it is easy to obtain the following sufficient condition for the existence ofk disjoint PC cycles.
Observation4. Let G be a colored Kn. If Δmon( )G ≤ n− 4 + 2k , then G contains k disjoint PC cycles of length at most 4.
Motivated by the above observations, our aim is to find a (best possible) positive function g k( ) (only depending on k) such that every colored complete graph G withΔmon( )G ≤ n− ( )g k containsk disjoint PC cycles. We conjecture that the following holds.
Conjecture 5. LetG be a colored Kn. IfΔmon( )G ≤n − 3 + 1k , thenG contains k disjoint PC cycles of length at most 4.
We confirm this conjecture for the case thatk = 2.
Theorem 6. LetG be a colored Kn. IfΔmon( )G ≤ n− 5, thenG contains two disjoint PC cycles of length at most 4.
We postpone the proof of Theorem 6 to Section 4. In Section 2, we give several additional results related to Conjecture 5. The proofs of these results can be found in Sections 5 and 6.
We continue here with some examples to discuss the tightness of the bounds in Conjecture 5. First of all, note that for a PC complete graph G on n= 3 − 1 vertices,k Δmon( ) = 1G ≤n − 3 + 2k , whereas it cannot have k disjoint PC cycles. This implies that the upper bound onΔmon( )G in Conjecture 5 would be best possible, in a weak sense: for eachk, this provides only one example. Whenk = 2, except for a PC K5, Example 1 also implies the tightness of the bound n − 5.
Example 1. Let G be a colored complete graph with V G( ) = { ,v v1 2, …, }v6. Decompose G −v1into two Hamilton cycles and color them byα and β, respectively. Color the edge
v v1 i withci for ∈i [2, 6]. ThenΔmon( ) = 6 − 4 = 2G , but G cannot contain two disjoint PC cycles.
Fork = 2, we have no other examples to support the tightness of the bound in Conjecture 5. Fork≥3, we cannot find other examples to support the bound in Conjecture 5 except for a PC K3 −1. It is not unlikely that the bound in Conjecture 5 can be improved for large n. The nextk example shows that for arbitrarily large n, we can construct a colored complete graph G on n vertices withΔmon( ) =G n − 3k
2 , but containing at mostk − 1 disjoint PC cycles.
Example 2. Given integers k ≥2 (k is even) and n ≥ 29k − 3, let G1≅K3 −3k with ≤ ≤
V G( 1) = { : 1vi i 3 − 3}k . Decompose G1 into 32k − 2 Hamilton cycles. Arbitrarily choose a direction for each Hamilton cycle. For alli j, ∈ [1, 3 − 3] and ≠k i j, color the edge v vi jwith a colorcjif and only if vjis the successor of viin one of the Hamilton cycles. LetG2≅ Kn−3 +3k withV G( 2) = { : 1ui ≤ ≤i n− 3 + 3}k and col G( 2) = { }α . Let G be an edge‐colored Kn constructed by joining G1 and G2 such that col v u( i n−3 +3k ) =β for all
∈
i [1, 3 − 3] and col v uk ( i j) =ci for all i j, with1≤ ≤i 3 − 3 andk 1≤ ≤j n− 3 + 2.k ThenΔmon( ) =G n− 3k
2 , but G contains at mostk − 1 disjoint PC cycles.
Since cycles in edge‐colored graphs are closely related to cycles in digraphs, here we naturally think of disjoint cycles in tournaments. In fact, Bang‐Jensen et al [3] proved that for every ϵ > 0, when k is large enough, every tournament with minimum out‐degree at least
(
32 +ϵ k)
containsk disjoint cycles. And the linear factor32 is better than the factor 2 that was conjectured by Bermond and Thomassen [5] in digraphs. In light of the close relationship between PC cycles in colored complete graphs and cycles in multipartite tournaments that we are going to discuss later, this could serve as supporting evidence that maybe the bound in Conjecture 5 can be improved when n is sufficiently large.
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A D D I T I O N A L R E S U L T S R E L A T E D T O C O N J E C T U R E 5
For the case thatk≥ 3, our first additional result implies the existence of k disjoint PC cycles if there exists a vertex in G that is not contained in any PC cycle.
Theorem 7. LetG be a colored Kn. IfΔmon( )G ≤n − 3 + 1k , then eitherG contains k disjoint PC cycles of length at most 4, or each vertex ofG is contained in a PC C3 orC4. Under some specific conditions, the bound forΔmon( )G in Theorem 7 can be improved to n− 2 .k
Theorem 8. Let G be a colored Kn satisfying Δmon( )G ≤ n− 2k. If G has a Gallai partition,1then eitherG contains k disjoint PC cycles of length at most 4, or each vertex of G is contained in a PCC3 orC4.
With the same upper bound onΔmon( )G , we can prove the following closely related result.
Theorem 9. LetG be a colored KnsatisfyingΔmon( )G ≤ n− 2k. Then eitherG contains k disjoint PC cycles of length at most 4, or each vertex ofG with color degree at most 3 is contained in a PCC3orC4.
Using some transformation techniques that we are going to specify later, it turns out that the results of Theorems 7 to 9 are closely related to a problem on disjoint cycles in multipartite tournaments. In 1981, Bermond and Thomassen posed the following conjecture on the existence of k disjoint cycles in digraphs. Here, δ D+( ) denotes the minimum out‐degree of the digraph D.
Conjecture 10 (Bermond and Thomassen [5]). Let D be a digraph. If δ D+( )≥ 2 − 1k , then D contains k disjoint cycles.
This conjecture has been confirmed for tournaments [3] and for bipartite tournaments [1] (for other progress on this conjecture, we refer to the introductory sections in [1,3]). We can state an equivalent of Conjecture 10 in terms of disjoint PC cycles when D is a multipartite tournament, using Theorem 11. We first define two classes of graphs. For convenience, we say a vertex in a colored graph is bad if there is no PC cycle passing through this vertex.
Definition 1. Letk ≥1,ℓ ≥ 2, f k( )≥ 2 − 1, andk I⊆ { :a a ≥3,a∈ . Define}
ℓ ℓ ≥
∈
{
}
I f k MT MT δ MT f k
i I
( , ( ), ) = is an ‐partite tournament with ( ) ( ) and containing no cycle of length
+ and ℓ ≤ ℓ ≤ ∣ ∣ ∈
{
}
I f k G G v d v G V G f k i I ( , ( ), ) =is a colored complete graph containing a bad vertex with ( ) , satisfying Δ ( ) ( ) − ( ) − 1 and containing no PC cycle of length
.
Gc mon
Theorem 11. ( , ( ), )I f k ℓ ≠ ∅ if and only if( , ( ), )I f k ℓ ≠ ∅. Furthermore, every digraph in( , ( ), )I f k ℓ hask disjoint cycles if and only if every graph in( , ( ), )I f k ℓ has k disjoint PC cycles.
Theorems 7 to 9, respectively, imply that every graph in ⋃ℓ≥2( , 3 − 2, )∅ k ℓ , ⋃ℓ≥2({3}, 2 − 1, )k ℓ , and⋃ℓ=2,3( , 2 − 1, )∅ k ℓ hask disjoint PC cycles. By directly using
Theorem 11, we immediately obtain the following three corollaries.2
Corollary 12. LetD be a multipartite tournament. Ifδ D+( )≥3 − 2k , thenD contains k disjoint cycles.
2During the process of writing this paper, we became aware of the fact that Bai and Li [2] have obtained Corollaries 12 to 14 in 2015 using techniques in
Corollary 13. Let D be a multipartite tournament containing no triangles. If ≥
δ D+( ) 2 − 1k , thenD contains k disjoint cycles.
Corollary 14. Let D be a 2‐partite or 3‐partite tournament. If δ D+( )≥ 2 − 1k , then D containsk disjoint cycles.
Finally, we present some structural properties of a possible minimum counterexample( , )G k to Conjecture 5. Here, a minimum counterexample( , ) satisfies that k is as small as possible,G k and subject to this, ∣V G( ) is as small as possible, and subject to this, ∣∣ col G( ) is as small as∣ possible.
Theorem 15. Let ( , ) be a minimum counterexample to Conjecture 5. Then theG k following statements hold.
(a) k ≥3;
(b) ∣col G( ) = 2 or 3;∣
(c) G contains no rainbow triangle; (d) G contains no monochromatic edge‐cut;
(e) for each setS⊆V G( ) with ∣ ∣ ≤S k − 1 and each vertexv∈ V G( )⧹S, there exists a PC C4inG−S containing v.
All the omitted proofs of the above results (except for the corollaries) can be found in Sections 4 to 6, but we start with some additional terminology and auxiliary lemmas in Section 3.
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T E R M I N O L O G Y A N D L E M M A S
Let G be a colored complete graph. A partition of G is a family of subsets V V1, 2, …, Vq ofV G( ) satisfying ⋃1≤ ≤i qVi =V G( ) andVi∩V =j ∅ for1≤i <j≤q (In the proofs, we sometimes allow that Viis an empty set.) For each partition V V1, 2, …, Vqof G and a vertexx∈ V G( ), we use Vx to denote the unique setVi (1≤ ≤i q)containing x. The following type of partition plays a key role in some of the proofs that follow. In this definition, the setsUi are supposed to be nonempty.
Definition 2 (Gallai partition). Let G be a colored Kn. A partition U U1, 2, …, Uq of G is called a Gallai partition if q≥2,∣⋃1≤i j q<≤ col U U( ,i j)∣ ≤2 and ∣col U U( ,i j) = 1∣ for
≤ i j≤ q
1 < .
The following result shows that Gallai partitions exist in colored complete graphs without a PC C3.
Lemma 16 (Gallai [8]). Let G be a colored Kn with n≥ 2. If G contains no rainbow triangles, thenG has a Gallai partition.
Let G be a colored Kn. Clearly, each monochromatic edge‐cut in G corresponds to a special Gallai partition of G. Actually, in the presence of a monochromatic edge‐cut, the degree conditionΔmon( )G ≤n − 2k easily implies the existence ofk disjoint PC cycles of length 4. In fact, we prove a slightly stronger result.
Lemma 17. Let G be a colored Kn satisfying Δmon( )G ≤ n− 2k. If G contains a monochromatic edge‐cut, then for each vertex ∈v V G( ), there is a set of k disjoint PC cycles of length 4 inG containing v.
Proof. Supposing that G contains a monochromatic edge‐cut, let V V1, 2be a partition of G with only one color (say red) appearing on the edges between V1and V2, and letv∈ V1. The conditionΔmon( )G ≤ n− 2kimplies thatk≥1 and that each vertex of V
1is joined to at least2 − 1 vertices of Vk 1with edges of colors distinct from red. Using induction onk, it is straightforward to see that this implies that there are k disjoint edges x x′ ,1 1 x x2 ′ , …,2 x xk ′kin G V[ ]1 with colors distinct from red and with v=x1. By symmetry, there are alsok disjoint edges y y1 ′ ,1 y y2 ′ , …,2 y yk ′kin G V[ ]2 with colors distinct from red. Thus,
≤ ≤ x x y y x i k
{ i ′ ′i i i i: 1 }is a set ofk disjoint PC cycles of length 4 containing v. □ Letk = 1 and let G be a colored complete graph. Lemma 17 implies that if G contains a bad vertex and satisfiesδ Gc( )≥ 2, then G has no monochromatic edge‐cut. In fact, a new partition result can be obtained in the presence of a bad vertex. Before delivering the result, we first give a description of this partition.
Definition 3 (v‐partition). Let G be a colored Kn and let v be a vertex of G. We say V V0, 1, …, Vpis a v‐partition of G if V V0, 1, …, Vp is a partition of G such that the following statements hold for some distinct colorsc c1, 2, …, cp∈col G( ).
(a) 2 ≤p≤d vc( ),v∈V0, and ∣ ∣ ≥V 1
i for0≤ ≤i p; (b) col V V( ,0 i) = { }ci for1 ≤ ≤i p;
(c) col V V( ,i j) ⊆ { , }c ci j for1≤i< j≤p;
(d) col G V( [ ])i ⊆ { }ci (ie, col G V( [ ]) = { }i ci when ∣ ∣ ≥Vi 2) for1≤ ≤i p.
Lemma 18. LetG be a colored Kn(n≥ 2) withδ Gc( )≥ 2. IfG contains a bad vertex v0, thenG admits a v0‐partition V V0, 1, …, Vp.
Proof. Since v0is a bad vertex andδ Gc( )≥ 2, by Lemma 17, G contains no monochromatic edge‐cut. Let N vc( ) =
{
c c, , …,c}
d v
0 1 2 c( )
0 and let Si= {v ∈V G( ):col vv( 0) = }ci for ≤ ≤i d v
1 c( )
0 . Since v0 is not contained in any PC triangle, we havecol S S( , )i j ⊆ { , }c ci j. Thus, the sets v{ },0 S S1, 2, …, Sd vc( )
0 form a partition of G satisfying (a), (b), and (c) of Definition 3.
Let V V0, 1, …, Vpbe a partition of G satisfying (a), (b), and (c), and with∣V0∣as large as possible. We will prove that this partition also satisfies (d). Suppose it does not. Then, without loss of generality, assume that there exist verticesx y, ∈ V1such thatcol xy( )≠ c1. For each vertexvj∈Vj (2≤ ≤j p), on the one hand, by (c), we havecol xv( j)∈{ , }c c1 j ; on the other hand, since v yxv v0 j 0 is not a PC cycle, we have col xv( j)∈{col xy c( ), }j . This forces that col xv( j) =cj. Similarly, we can prove that col yv( j) =cj. This implies that col x V( , j) =col y V( , j) = { }cj for all j with2 ≤ ≤j p. Now define
∈ ∃ ∈ ≠
T1= {x V1: y V1 s.t.col xy( ) c1}.
Then, col T V( ,1 j) = { }cj for 2≤ ≤j p. Let V′ =1 V T1⧹ 1 and V′ =0 V0∪ T1. Then, V′ ,0 V′ ,1 V2, …, Vp is a new partition of G. IfV ′1≠ ∅, then by the definition of T1, we have col V( ′ ,1 T1) = { }c1 andcol G V( [ ′])1 ⊆ { }c1. Thus, V′ ,0 V′ ,1 V2, …, Vpis a partition of G
satisfying (a), (b), and (c) with ∣V′ >0∣ ∣ ∣V0. This contradicts the choice of V V0, 1, …, Vp. If ∅
V ′ =1 , then p≥3 (otherwise, p = 2, T1=V1and the edges betweenV ′0and V2form a monochromatic edge‐cut of G, a contradiction). Thus, V′ ,0 V2, …, Vp is a partition of G satisfying (a), (b), and (c) with ∣V′0∣ >∣ ∣V0, a contradiction. So (d) also "13"holds. □ In the cases that a colored complete graph G does and does not admit a Gallai partition, respectively, the next two lemmas give extra structural results beyond the v‐partition.
Lemma 19. LetG be a colored Kn(n≥ 2) withδ Gc( ) ≥2. IfG has a Gallai partition and contains a bad vertex v0, thenG has a v0‐partition V V V0, 1, 2 with v0∈ V0 and a PC cycle xyzwx with x z, ∈V1and y w, ∈ V2.
Proof. By Lemma 17, G contains no monochromatic edge‐cut. Choose U U0, 1, …, Uqas a Gallai partition of G withv0∈U0. By Definition 2, there are at most two colors (say red andblue) between the sets{ : 0Ui ≤ ≤i q}and exactly one color (say red orblue) between each pair of distinct setsUiand Uj(0≤i< j≤q). Since there is no monochromatic edge‐ cut in G, for each ∈i [0, ], there existq s t, ∈[0, ] with ≠q i s i, ≠t, ands≠t such that col U U( ,i s) = {red} and col U U( ,i t) = {blue}. Let G′ =G −U0⧹{ }v0 (it is possible that U0= { }v0 andG′ =G). Thus,dGc′( ) = 2v0 and v0is also bad inG′ withδ Gc( ′)≥ 2. Hence, by Lemma 18,G′ has a v0‐partition V′ ,0 V′ , …,1 V′p with2 ≤p≤dGc′( )v0 and v0∈ V ′0. Recall thatdGc′( ) = 2v0 . We have p = 2. LetV0=V′0 ∪U0, V1=V′1, and V2=V′2. By the property of Gallai partitions, for each vertexu ∈ ∪iq=1Ui, we have col U u( 0, ) = {col v u( 0 )}. Note thatV ∪V ⊆ ∪i U
q i
1 2 =1 . We can easily see that V V V0, 1, 2is a v0‐partition of G. In this case, we are left to prove the existence of a specific PC C4. IfG V[ 1∪V2] contains a PC cycle, then it must be a PC C4 (because of Observation 1 and ∣col G V( [ 1∪V2]) = 2∣ ) with two vertices in V1(sayx z, ) and two vertices in V2(sayy w, ). Sincecol xz( )≠ col yw( ), this C4must bexyzwx. So, it is sufficient to prove thatG V[ 1∪V2]contains a PC cycle. Suppose the contrary. Then, by Theorem 2, we may assume that there exists a vertexx∈V1joined to all the other vertices inV1∪V2with edges of the same color. If this unique color is c1, then dGc( ) = 1x , a contradiction; otherwise, this unique color is c2. This forces that V1= { }x . Then, the edges betweenV0∪{ }x and V2 form a monochromatic edge‐cut of G,
again a“21”contradiction. □
Lemma 20. LetG be a colored Kn (n≥ 2) and letv0 be a bad vertex ofG. If G has no Gallai partition and has a v0‐partition V V0, 1, …, Vp, thenG contains a rainbow triangle xyzx such that Vx,Vy, andVz are three distinct sets withV0∉{ ,V V Vx y, z}.
Proof. Supposing that G contains no Gallai partition, by Lemma 16, G must contain a rainbow triangle. Let V V0, 1, …, Vp be a v0‐partition of G. Let G′ =G−V0⧹{ }v0 (it is possible that V0= { }v0 andG′ =G). Then,{ },v0 V1, …, Vpis a v0‐partition of G′. Now we are left to prove the existence of a specific rainbow triangle inG′. Assume that G′ contains a rainbow triangle xyzx. Since v0 is bad and ∣col G V( [ i ∪Vj])∣ ≤2 for 1≤i< j≤p, the verticesx y z, , must come from different sets in V{ , …,1 Vp}. So it is sufficient to prove that G′ contains a rainbow triangle. Suppose the contrary. Since ∣V G( ′)∣ ≥p+ 1≥3, by Lemma 16,G′ has a Gallai partitionU U U0, 1, 2, …, U qq ( ≥1). Without loss of generality, assume that v0∈U0. On the one hand, by the property of v0‐partitions, for each pair of vertices v∈V0 andu∈ ∪ip=1Vi, we have col vu( ) =col v u( 0 ); on the other hand, by the
property of Gallai partitions, for each pair of vertices v∈U0 andu ∈ ∪iq=1Ui, we have col vu( ) =col v u( 0 ). Note that ∪i U ⊆ ∪ V
q
i i
p i
=1 =1 . So for each pair of vertices v∈ U0∪V0 and u∈ ∪ip=1Ui, we have col vu( ) =col v u( 0 ). Thus, U0∪V U U0, 1, 2, …, Uq is a Gallai partition of G, which contradicts that G has no Gallai partition. □ We now have all the necessary ingredients to prove our main theorem and the additional results. In Section 4, we present our proof of Theorem 6.
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P R O O F O F T H E O R E M 6
Proof. By contradiction. Let G be a colored complete graph satisfyingΔmon( )G ≤n − 5 but containing no two disjoint PC cycles. SinceΔmon( )G ≥1, we haven≥6. If G contains a rainbow triangle uvwu, then by deleting verticesu v, , and w from G, we obtain a graph G′ with ∣V G( ′) =∣ n− 3≥ 3 andΔmon( ′)G ≤n− 5 = ( − 3) − 2n . So, by Observation 3, G′ contains a PC cycleCof length 3 or 4. Thus, the cycles uvwu andCform two disjoint PC cycles of length at most 4, a contradiction. Hence G contains no rainbow triangles, and, by Lemma 16, G has a Gallai partition. Let U U1, 2, …, Uq be a Gallai partition of G withqas small as possible. By Lemma 17, G contains no monochromatic edge‐cut. So, we
haveq≥4. Assume that the two colors appearing betweenUiandUj (1≤ i<j≤ q)are red andblue.
We proceed by proving six claims.
Claim 1. There exists a PC C4 in G with vertices from distinct sets of U U1, 2, …, Uq. Proof. Construct an auxiliary colored complete graphH withV H( ) = { , , …,x x1 2 xq}and for1 ≤i< j≤q, color the edge x xi jwith the color that appears on the edges betweenUi and Uj. Since G contains no monochromatic edge‐cut, col H( ) = {red blue, } and col x H( , − ) = {x red blue, } for each vertex x∈V H( ). Thus, by Observation 3, H
contains a PC C4, which corresponds to a PC C4 in G with vertices from different sets
of U U1, 2, …, Uq. □
Without loss of generality, assume that the PC C4in Claim 1 is C*=v v v v v1 2 3 4 1withvi∈Ui for1 ≤ ≤i 4, and satisfying that col v v( 1 2) =col v v( 3 4) =red and col v v( 2 3) =col v v( 1 4) =blue (see Figure 1). Let G′ =G −V C( * . Since ∣) V G( ′)∣ ≥n− 4≥ 2, G′ is nonempty. If
≤ ∣ ∣
G V G
Δmon( ′) ( ′) − 2, then, by Observation 3, G′ contains a PC C4. Combining this PC cycle with v v v v v1 2 3 4 1, we get two disjoint PC cycles of length 4, a contradiction. So, there exists a vertexv ∈V G( ′) with dGc′( ) = 1v (see Figure 1). Define
∈
{
}
S1= v V G( ′):dGc′( ) = 1 .v
Clearly, there is only one color incol S G( , ′ −1 S1)∪col G S( [ ])1 . We assert that this color must be red orblue. Suppose not. Then, by the definition of Gallai partition, V G( ′) is a subset ofUifor some i with 1≤ ≤i q. Let vj be a vertex in Uj for some j with 1≤ ≤j q and j≠i. Then, the unique color in col U U( ,i j)appears at least ∣V G( ′) =∣ n− 4 times at vj. This contradicts that
≤
G n
Δmon( ) − 5. Now, without loss of generality, assume that col S G( , ′ −S)∪
1 1
col G S( [ ]) = {1 red}.
Claim 2. For each vertexv∈S1,v∉ ∪1≤ ≤i 4Ui, and col v C( , *) = {blue}.
Proof. Let v be an arbitrary vertex of S1. By the assumption that col S( ,1 ∪
G′ −S1) col G S( [ ]) = {1 red}, we have col v G( , ′ − ) = {v red}. Then, col vv( i) ≠red for ≤ ≤i
1 4 (otherwise, the color red would appear more than n − 5 times at v, a contradiction). We further assert that v∉ ∪1≤ ≤i 4Ui. Suppose this is not the case. Then,
∈
v Uifor some1≤ ≤i 4, and col vv( i+1) =redorcol vv( i−1) =red(where the indices are taken module 4), a contradiction. This implies thatv ∉ ∪1≤ ≤i 4Ui and col v C( , *) = {blue}
(see Figure 1). □
Claim 3. Ui= { }vi for1≤ ≤i 4.
Proof. Claim 2 shows thatS1∩ ∪{ 1≤ ≤i 4Ui} =∅. We are left to prove thatu ∉ ∪1≤ ≤i 4Ui for each vertexu∈ V G( ′)⧹S1. Note that for each vertexu∈ V G( ′)⧹S1 and any vertex
∈
v S1, we havecol vu( ) =red∉ col v C( , * . This implies that) u∉ ∪1≤ ≤i 4Ui. □ Now, for convenience, we call a cycle special if it is a PC cycle and its vertices come from different sets of U U1, 2, …, Uq. We say a vertexz∈V G( )⧹V C( ) is a companion vertex of a special cycleCifzis joined toCwith colorblue(red) and joined to other vertices with color red (blue). By Claims 2 and 3, we know that
(a) each special cycle of length 4 in G has a companion vertex; (b) if a vertexvi∈Uiis contained in a special cycle, then Ui= { }vi.
Claim 4. ∣ ∣ ≤S1 3, and for each vertex vi (1≤ ≤i 4), there exist two distinct vertices
∈ ⧹ ∪
x yi, i V G( ) ( (V C*) S1) such that col v x( i i) =col v y( i i) =red.
Proof. Suppose that ∣ ∣ ≥S1 4. Let x y z w, , , be four distinct vertices in S1. Then, xyv v x1 2 and zwv v z3 4 are two disjoint PC cycles, a contradiction. So, we have ∣ ∣ ≤S1 3. For eachi with1≤ ≤i 4, by Claim 3 and the definition of Gallai partition, we know that all the edges incident with viare colored in red orblue. Since each color appears at most n − 5 times at vi, we know that both red andblueappear at least 4 times at vi. Note that there are at most 2
vertices inV C( *)∪S1joined to viby an edge with color red. Hence, there exist another two vertices x yi, i ∈V G( ) ( (⧹ V C*)∪ S1) joined to viby an edge with color red. □ Let G″ =G′ −S1. By Claim 4, we have ∣V G( ″)∣ ≥2. IfΔmon( ″)G ≤ ∣V G( ″) − 2∣ , then, by Observation 3,G″ contains a PC C4. Combining this cycle withC*, we obtain two disjoint PC C4’s, a contradiction. Thus, there exists a vertex ∈u V G( ″) such that dGc″( ) = 1u (see Figure 1). Define
∈
{
}
S2= u V G( ″):dGc″( ) = 1 .u
Claim 5. For each vertex u∈S col u G2, ( , ″ − ) = {u blue U}, u∩S1=∅, and col u C( , *) = {red}.
Proof. Let u be a vertex in S2. Then dGc″( ) = 1u .
Suppose that col u G( , ″ − ) = {u red}. Then, col u G( , ′ − ) =u col u G( , ″ − )u ∪ col u S( , 1) = {red}. This implies that u∈ S1, a contradiction. Suppose that the unique color incol u G( , ″ − ) is neither red noru blue. Then, by Claim 3 and the definition of Gallai partition, we haveV G( ″) ⊆ Uj for some j with5 ≤ ≤j q. By Claim 4, there are vertices x x x x1, ,2 3, 4 in V G( ″) such that col v x( i i) =red for all i with 1≤ ≤i 4. This implies that col v U( ,i j) = {red}for alli with1≤ ≤i 4. Let
T1= { ,v v v v1 2, 3, },4 T2=S1, and T3=V G( ″).
It is easy to check that col T T( ,1 2) = {blue col T T}, ( ,2 3) = {red}, and col T T( ,1 3) = {red}. Thus, the edges betweenT1∪T2 and T3form a red edge‐cut of G, a contradiction. So, we have col u G( , ″ − ) = {u blue}.
Suppose that there exists a vertex v ∈S1 such that v ∈Uu. Then, u∈Uv and col u C( , *) =col v C( , *) = {blue}. Thus, col u G( , −S1) =col u G( , ″ − )u∪ col u C( , *) =
blue
{ }. This implies that the colorblue appears at leastn− 1 −∣S1∣ ≥n− 4 times at the vertex u (by Claim 4), a contradiction. Thus, we haveUu∩S =1 ∅.
Now we need to prove that col uv( i) = {red} for alli with1≤ ≤i 4. Suppose, to the contrary, that there exists a vertex (say v1) onC* such that col uv( 1) =blue. Then, choose a vertexv ∈S1. Thus, the cycle C=vuv v v1 2 is a special C4(see Figure 2). Recall that each special cycle of length 4 has a companion vertex. Letz be a companion vertex ofC. For verticesx∈ V G( ″) −uandy∈S1−v, we havecol xu( )≠ col xv( ) andcol yv( 1)≠col yv( ). This implies that z∉V G( ″)∪ S1. Thus, z is either v3 or v4. If z=v3, then col z C( , ) = {col v v( 3 2)} = {blue}. By the definition of z, we know that col z V G( , ( ′) −u− ) = {v red}. Note that col z S( , 1) =col v S( ,3 1) = {blue}. This forces that S1= { }v . Now, for each vertex x∈V G( ) \ { , , , }v v v u2 4 , we have col ux( ) =blue. The colorblueappears at leastn − 4 times at u, a contradiction. Soz≠ v3. Similarly, we can prove that z≠ v4. Thus, there is no choice for z, a contradiction. This implies that
col uv( i) =redfor all1≤ ≤i 4. □
Claim 6. V G( ″)⧹S2≠ ∅, and there exists a vertex w∈V G( ″)⧹S2 such that col w C( , *) = {red blue, }andw∉ Uv∪ Uu for any verticesv∈ S1andu ∈S2.
Proof. IfV G( ″)⧹S2=∅, then the edges between S2and G−S2form a red edge‐cut of G, a contradiction. So, we haveV G( ″)⧹S2≠ ∅. Suppose that for each vertexw ∈V G( ″)⧹S2, we have col wv( i) =col wv( j) for all 1≤ i<j≤ 4. Recall that col S C( ,1 *) = {blue} and col S C( ,2 *) = {red} (by Claim 5). We have col xv( i) =col xv( j) for each vertex
∈ ⧹
x V G( ) V C( * and) 1≤ i<j≤ 4. Thus,
v v v v U U U { ,1 2, 3, },4 5, 6, …, q
is also a Gallai partition of G. This contradicts thatqis as small as possible. Thus, we can choose a vertex w ∈V G( ″)⧹S2 such that col w C( , *) = {red blue, }. Since col S C( ,1 *) = {blue} and col S C( ,2 *) = {red}, by the definition of Gallai partition,
∉ ∪
w Uv Uufor any verticesv∈ S1 andu∈S2.
Since col w C( , *) = {red blue, }, without loss of generality, assume thatcol wv( 1) =red. Choose verticesv ∈S1andu∈S2. Then, the cycle C=vuwv v1 is a special cycle of length 4 (see Figure 3). Letzbe a companion vertex ofC. Sincecol vx( )≠col ux( ) for each vertex x inG− (S1∪{ })u , we have z∈ S1−v. However, for each vertex y∈S1−v, we have col yv( 1) =blue and col yu( ) =red. Thus, z∉ S1−v. So there is no choice for z, a
contradiction. This completes the proof of Theorem 6. □
5
|
P R O O F S O F T H E O R E M S 7 , 8 , 9 , A N D 1 5
By Observation 1, the existence ofk disjoint PC cycles is equivalent to the existence of k disjoint PC C3’s or C4s. In this section, for convenience, we also use the term short PC cycle(s) instead of PC cycle(s) of length at most 4.
Proof of Theorem 7. By contradiction. Let G be a colored Kn. We say (G k, ) is a counterexample to Theorem 7 ifΔmon( )G ≤n − 3 + 1k , but there are nok disjoint short PC cycles in G and not every vertex of G is contained in a short PC cycle. Let (G k, ) be a counterexample to Theorem 7 withk as small as possible. By Observation 3 and Theorem 6, we know thatk≥ 3. If G contains a rainbow triangle xyzx, then let H=G− { , , }.x y z Then, Δmon( )H ≤Δmon( ) =G n − 3 − 3( − 1) + 1k . Hence, by the choice of ( , ),G k H either containsk − 1 disjoint short PC cycles, or each vertex ofH is contained in a short PC cycle. This in turn implies that either G containsk disjoint short PC cycles, or each vertex of G is contained in a short PC cycle, a contradiction. Thus, G contains no rainbow triangles and, due to Lemma 16, has a Gallai partition. Note that Δmon( )G ≤ n− 3 +k
n k
1 < − 2 . By Theorem 8, G either contains k disjoint short PC cycles, or each vertex is contained in a short PC cycle. This completes the proof. □ Proof of Theorem8. By contradiction. Let (G k, ) be a counterexample to Theorem 8 with k as small as possible. Then G contains a bad vertex v0. By Observation 3, we havek≥2. By Lemma 17, G contains no monochromatic edge‐cut. Since G contains a Gallai partition, by Lemma 19, G admits a v0‐partition V V V0, 1, 2 (see Figure 4) such that
∈ ⊆
v V col V V c col V V c
col V V c c col G V c col G V c
, ( , ) = { }, ( , ) = { }, ( , ) { , }, ( [ ]) = { }, ( [ ]) = { }
0 0 0 1 1 0 2 2
1 2 1 2 1 1 2 2
and G contains a PC cycle xyzwx with
∈ ∈
x z, V1, y w, V2. This implies that
∣ ∣ ≥V1 2 and ∣ ∣ ≥V2 2.
Without loss of generality, assume that col xy( ) =col zw( ) =c1andcol xw( ) =col zy( ) =c2.
SincedGc1( )x ≤ n− 2k anddGc2( )y ≤ n− 2k, we have
∣ ∣V0 + (∣ ∣Vi − 1) + 1≤n − 2k for i= 1, 2. Thus,
≤ ∣ ∣ ≤V n k i
2 i − 2 − 1 for = 1, 2.
F I G U R E 3 col wv( 1) =red[Color figure can be viewed at wileyonlinelibrary.com]
Let H=G− { , , , }. We will show thatx y z w Δmon( )H ≤n− 2 − 2k . For each vertex
∈ ⧹
v1 V1 { , }x z , by the partition, we know that col v G( ,1 −v1) ⊆ { ,c c1 2},
≤ ≤
dHc1( )v1 dGc1( ) − 2v1 n− 2 − 2k , and dHc2( )v1 ≤ ∣ ∣V2 − 2 < n− 2 − 2k . Similarly,
for each vertex v2∈V2⧹{ , }y w, we have col v G( ,2 −v2) ⊆ { ,c c1 2}, dHc2( )v2 ≤dGc2( ) −v2
≤ n k
2 − 2 − 2, anddHc1( )v2 ≤ ∣ ∣V1 − 2 < n− 2 − 2k . For each vertexu ∈V0, we have
≤ ≤
dHc1( )u dGc1( ) − 2u n− 2 − 2k ,dHc2( )u ≤ dGc2( ) − 2u ≤n − 2 − 2k , and dHc( )u ≤
∣ ∣V0 − 1≤n− 2 −k ∣ ∣V1 − 1 <n− 2 − 2k for each colorc∈col G( ) { ,⧹ c c1 2}. This implies that Δmon( )H ≤n − 2 − 2 =k ∣V H( ) − 2( − 1)∣ k . Note that ( , ) is a counterexampleG k withk as small as possible, and that the vertex v0is also bad inH. We know thatH contains k − 1 disjoint short PC cycles. Together with the PC cycle xyzwx, there exist k disjoint short PC cycles in G, a contradiction. This completes the proof of Theorem 8. □ Proof of Theorem9. By contradiction. Let( , ) be a counterexample to Theorem 9 withG k k as small as possible. Then G contains a bad vertex v0withd vc( )0 ≤3. By Observation 3,
≥
k 2. By Theorem 8, G admits no Gallai partition. Furthermore, by Lemmas 18 and 20, G has a v0‐partition V V0, 1, …, Vpand a rainbow trianglexyzx such that V V Vx, y, z are distinct sets with V0∉{ ,V V Vx y, z}. So we have 3≤p≤d vc( )0 ≤3. This forces p = 3 (see Figure 5). Without loss of generality, assume that
∈ ∈ ∈
x V1, y V2, z V3, and
col xy( ) =c1, col yz( ) =c2, col zx( ) = c3.
Since colors c1, c2, and c3appear at mostn− 2 times at x, y, andk z, respectively, we have ∣ ∣V0 + (∣ ∣Vi − 1) + 1≤ n− 2k for i= 1, 2, 3.
Thus
≤ ∣ ∣ ≤V n k i
1 i − 2 − 1 for = 1, 2, 3. Let H=G− { , , }. We will show thatx y z Δmon( )H ≤n − 2 − 1k .
For each vertexv1∈ V1⧹{ }x , by the partition, we know thatcol v G( ,1 −v1)⊆ c c c{ ,1 2, }3,
≤ ≤
dHc1( )v1 dGc1( ) − 1v1 n− 2 − 1k , and dHci( )v1 ≤ ∣ ∣Vi − 1 <n− 2 − 1k for i = 2, 3.
Similarly, for vertices v2∈V2⧹{ }y and v3∈ V3⧹{ }z , we have col v G( ,2 −v2)∪ col v G( ,3 −v3) = { ,c c c1 2, }3 and dHci( )vj ≤ n− 2 − 1k for i = 1, 2, 3 and j = 2, 3. For each vertex u ∈V0, we have dHci( )u ≤dGci( ) − 1u ≤ n− 2 − 1k for i = 1, 2, 3, and
≤ ∣ ∣
dHc( )u V0 − 1 <n− 2 − 1k for each colorc∈ col G( ) { ,⧹ c c c1 2, }3. This implies that
≤ ∣ ∣
H n k V H k
Δmon( ) − 2 − 1 = ( ) − 2( − 1). By the choice of( , ), and since v0G k is bad, we conclude thatH containsk − 1 disjoint short PC cycles. Together with the PC cycle xyzx, there exist k disjoint short PC cycles in G, a contradiction. This completes the proof
of Theorem 9. □
Proof of Theorem15. By Observation 3 and Theorem 6, we havek ≥3. If G contains a rainbow trianglexyzx, then one easily checks that G − { , , } is a smaller counterexamplex y z to Conjecture 5, a contradiction. So, G contains no rainbow triangles, and thus has a Gallai partition U U1, 2, …, Uqby Lemma 16. By Lemma 17, G contains no monochromatic edge‐cut. So q≥4, and we can assume that the two colors appearing between the parts in this Gallai partition are red andblue. SinceΔmon( )G ≤n− 3 + 1k , by the definition of Gallai partition, we have ∣ ∣ ≤Ui n− 3 + 1k for all i with 1≤ ≤i q. This implies that ∑c col G c red blue∈ ( ), ≠ , dGc( )vi ≤ ∣ ∣Ui − 1 < n− 3 + 1k for each vi∈ Ui (1≤ ≤i q). If ∣col G( )∣ ≠2 or 3, then ∣col G( )∣ ≥4. In this case, let H be the colored graph obtained from G by recoloring all the edges which are neither red norbluein G with the colorgreen. Clearly,H contains nok disjoint short PC cycles (otherwise, there exist k disjoint short PC cycles in G), andΔmon( )H ≤ ∣V H( ) − 3 + 1∣ k . So,( , ) is a counterexample to ConjectureH k 5 with ∣V H( ) =∣ ∣V G( ) and ∣∣ col H( ) <∣ ∣col G( ) , a contradiction. Hence, we conclude that∣ ∣col G( ) = 2 or 3. This completes the proof of (a), (b), (c), and (d) of Theorem 15.∣
Finally, we prove (e) of Theorem 15, by contradiction. Suppose to the contrary, that there exists a setS⊆ V G( ) with ∣ ∣ ≤S k − 1, and a vertexv0∈ V G( )⧹S such that v0is a bad vertex inG− . LetS H=G− . Then,S
≤ ∣ ∣ ∣ ∣ ≤ ∣ ∣ H G n k n S k S k V H k Δ ( ) Δ ( ) = − 3 + 1 = ( − ) − 2 + ( − + 1) ( ) − 2 . mon mon
By Theorem 8 and the fact that v0is bad inH, the colored complete graphH containsk disjoint short PC cycles, which are also contained in G, a contradiction. This completes
the proof of Theorem 15. □
6
|
P R O O F O F T H E O R E M 1 1
Before delivering the proof of Theorem 11, we first give the following two constructions. Construction 1. Let MT1∈ ( , ( ), )I f k ℓ with partite sets V V1, 2, …, Vℓ. Define a colored complete graph G1 with V G( 1) =⋃1≤ ≤ℓi Vi ∪{ }v0 , for 1≤ ≤ ≤ ℓi j ,vi≠ vj,
∈ ∈ vi V vi, j V col v vj, ( 0 i) =ci and ∈ col v v c v v A MT c ( ) = if ( ), otherwise. i j j i j i ⎧ ⎨ ⎩
Construction 2. LetG2∈ ( , ( ), )I f k ℓ and letv0be a bad vertex inG2withd vc( )0 ≤ ℓ. ThenG2contains no monochromatic edge‐cut (otherwise, it is easy to see that every vertex in G2is contained in a PC cycle). By Lemma 18,G2admits av0‐partition V V V0, 1, 2, …, Vpsuch thatv0∈V0andp≤ d vc( )0 ≤ ℓ. Then we define ap‐partite tournament MT with vertex set
∪ ∪ ⋯∪
V MT( ) =V1 V2 Vp and arc set ≠
A MT( ) = { :xy Vx Vy, col xy( ) =col v y( 0 )}.
Define anℓ‐partite tournament MT2. If p = , we take MTℓ 2=MT; otherwise, let
∪ ℓ
V MT( 2) =V MT( ) { ,u u1 2, …,u −p} and
∪ ∈ ≤ ≤ ℓ ∪ ≤ ≤ ℓ
A MT( 2) =A MT( ) {u x xi : V MT( ), 1 i − }p {u uj i: 1 i< j − }.p
Proof of Theorem11. Let MT1and G2 be arbitrarily chosen graphs from( , ( ), )I f k ℓ and( , ( ), )I f k ℓ, respectively. Define G1 and MT2 by Constructions 1 and 2. To prove Theorem 11, it is sufficient to prove the following statements: (i)G1∈( , ( ), )I f k ℓ , and if G1 contains k disjoint PC cycles, then MT1 contains k disjoint cycles; (ii)
∈ ℓ
MT2 ( , ( ), )I f k , and if MT2 containsk disjoint cycles, then G2 containsk disjoint PC cycles.
(i) By Construction 1, we havecol G V( 1[ ])i ⊆ { }ci , that is, col G V( 1[ ]) = { }i ci when ∣ ∣ ≥Vi 2 for all i with 1≤ ≤ ℓi . Let n=∣V G( 1)∣. Then n=∑1≤ ≤ℓi ∣ ∣Vi + 1. For a vertex
∈ ≤ ≤ ℓ
vi Vi (1 i ), denote by NMT+1( )vi the set of out‐neighbors of vi in MT1. Since ∣NMT+1( ) =vi ∣ dMT+1( )vi ≥f k( ) andNMT+1( )vi ∩Vi=∅, we have∣ ∣ ≤Vi n− ( ) − 1f k . Note that each vertex in NMT+1( )vi is joined to viby an edge with color distinct fromciin G1. So, the color ci appears at most n− ( ) − 1 times at vf k i, and any color c jj ( ≠i) may appear at most ∣ ∣ ≤Vj n− ( ) − 1f k times at vi. For the vertex v0, each color appears at most ∣ ∣ ≤Vj n− ( ) − 1f k times at v0. Thus, we haveΔmon(G1)≤ n− ( ) − 1f k . Claim 1. v0is a bad vertex in G1and each edgexy is not contained in any PC cycle in G1 for x y, ∈ Vi (1≤ ≤ ℓi ).
Proof. Suppose thatCis a PC cycle in G1containing v0. Orient the edges ofCin one of the two directions alongC. Choose a vertexu ∈V C( ) { }⧹ v0, and assume thatu∈ Vi for someiwith1≤ ≤ ℓi . Then, we obtaincol u u( − ) =ciandcol u u( + ) =ci, by following the pathsv Cu u0 ⃗ − and
←
v C u u0 + , respectively. (Here,u+andu−denote the immediate successor and predecessor of u onCin the direction specified by the orientation ofC, respectively, andC and⃗ ←C denote the traversal ofC in the direction of the orientation, and in the opposite direction, respectively.) Thuscol u u( + ) =col u u( − ), a contradiction. Similarly, we can prove that xy is not contained in any PC cycles forx y, ∈Vi (1≤ ≤ ℓi ).
Claim 1 implies that each PC cycle in G1corresponds to a cycle in MT1. So G1contains no PC cycle of length i∈ I and v0 is a bad vertex in G1 with dGc1( )v0 ≤ ℓ. In summary,
∈ ℓ
G1 ( , ( ), )I f k , and if G1 containsk disjoint PC cycles, then MT1 containsk disjoint cycle. Hence, i( ) holds.
(i) Let V V V0, 1, 2, …, Vp be the v0‐partition of G2 in Construction 2. Note that for each vertex
∈ ≤ ≤
vi Vi (1 i p), there are at least f k( ) vertices in G2−V0 joined to vi by edges with colors different from ci. This implies that δ MT+( 2)≥f k( ). Note that every cycle in MT2 corresponds to a PC cycle in G2. So MT2 contains no cycle of lengthi ∈I . In summary,
∈ ℓ
MT2 ( , ( ), )I f k and if MT2 containsk disjoint cycles, then G2containsk disjoint PC cycles. Hence (ii) holds. This completes the proof of Theorem 11. ◻
A C K N O W L E D G M E N T S
The authors are grateful to the referees for their valuable comments and suggestions. The research is supported by CSC (No. 201506290097), NSFC (Nos. 11671320 and 11901459), and the Fundamental Research Funds for the Central Universities of China (Nos. 31020180QD124 and 3102019GHJD003), and partially by the group of Formal Methods and Tools when the first author visited University of Twente.
O R C I D
Ruonan Li http://orcid.org/0000-0001-5419-1533
Hajo Broersma http://orcid.org/0000-0002-4678-3210
Shenggui Zhang http://orcid.org/0000-0002-9596-0826
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How to cite this article: Li R, Broersma H, Zhang S. Vertex‐disjoint properly
edge‐colored cycles in edge‐colored complete graphs. J Graph Theory. 2020;94:476–493.
