λ-backbone colorings along pairwise disjoint stars and matchings

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Contents lists available atScienceDirect

Discrete Mathematics

journal homepage:www.elsevier.com/locate/disc

λ

-backbone colorings along pairwise disjoint stars and matchings

H.J. Broersma

a,∗

_{, J. Fujisawa}

b

_{, L. Marchal}

c

_{, D. Paulusma}

a

_{, A.N.M. Salman}

d

_{, K. Yoshimoto}

e

a_{Department of Computer Science, Durham University, South Road, DH1 3LE, Durham, United Kingdom} b_{Department of Computer Science, Nihon University, Sakurajosui 3-25-40, Setagaya-Ku, Tokyo 156-8550, Japan} c_{Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands}

d_{Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung,} Jalan Ganesa 10 Bandung 40132, Indonesia

e_{Department of Mathematics, College of Science and Technology, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8308, Japan}

a r t i c l e i n f o Article history:

Received 4 August 2006 Accepted 1 April 2008 Available online 14 May 2008 Keywords:

λ-backbone coloring

λ-backbone coloring number Star

Matching

a b s t r a c t

Given an integerλ ≥2, a graphG=(V,E)and a spanning subgraphHofG(the backbone of G), aλ-backbone coloring of(G,H)is a proper vertex coloringV→ {1,2, . . .}ofG, in which the colors assigned to adjacent vertices inHdiffer by at leastλ. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backboneSofGthe minimum number`for which aλ-backbone coloring of(G,S) with colors in{1, . . . , `}exists can roughly differ by a multiplicative factor of at most 2−1

λ

from the chromatic numberχ(G). For the special case of matching backbones this factor is roughly 2− 2

λ+1. We also show that the computational complexity of the problem “Given a graphGwith a star backboneS, and an integer`, is there aλ-backbone coloring of(G,S)with colors in{1, . . . , `}?” jumps from polynomially solvable toNP-complete between` = λ+1 and` = λ +2 (the case` = λ +2 is evenNP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

1. Introduction

In [7] backbone colorings are introduced, motivated and put into a general framework of coloring problems related to frequency assignment.

Graphs are used to model the topology and interference between transmitters (receivers, base stations, sensors): the vertices represent the transmitters; two vertices are adjacent if the corresponding transmitters are so close (or so strong) that they are likely to interfere if they broadcast on the same or ‘similar’ frequency channels. The problem is to assign the frequency channels in an economical way to the transmitters in such a way that interference is kept at an ‘acceptable level’. This has led to various types of coloring problems in graphs, depending on different ways to model the level of interference, the notion of similar frequency channels, and the definition of acceptable level of interference (see, e.g., [16,20]). Although new technologies have led to different ways of avoiding interference between powerful transmitters, such as base stations for mobile telephones, the above coloring problems still apply to less powerful transmitters, such as those appearing in sensor networks.

We refer the reader to [6,7] for an overview of related research, but we repeat the general framework and some of the related research here for convenience and background.

∗_{Corresponding author.}

E-mail addresses:hajo.broersma@durham.ac.uk(H.J. Broersma),fujisawa@cs.chs.nihon-u.ac.jp(J. Fujisawa),b.marchal@ke.unimaas.nl(L. Marchal),

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Given two graphsG1andG2with the property thatG1is a spanning subgraph ofG2, one considers the following type of

coloring problems: Determine a coloring of (G1and)G2that satisfies certain restrictions of type 1 inG1, and restrictions

of type 2 inG2.

Many known coloring problems fit into this general framework. We mention some of them here explicitly, without giving details. First of all suppose thatG2=G21, i.e.G2is obtained fromG1by adding edges between all pairs of vertices that are at

distance 2 inG1. If one just asks for a proper vertex coloring ofG2(andG1), this is known as the distance-2-coloring problem.

Much of the research has been concentrated on the case thatG1is a planar graph. We refer to [1,4,5,18,21] for more details.

In some versions of this problem one puts the additional restriction onG1that the colors should be sufficiently separated, in

order to model practical frequency assignment problems in which interference should be kept at an acceptable level. One way to model this is to use positive integers for the colors (modeling certain frequency channels) and to ask for a coloring ofG1andG2such that the colors on adjacent vertices inG2 are different, whereas they differ by at least 2 on adjacent

vertices inG₁. A closely related variant is known as the radio coloring problem and has been studied (under various names) in [2,9–13,19]. A third variant is known as the radio labeling problem and models a practical setting in which all assigned frequency channels should be distinct, with the additional restriction that adjacent transmitters should use sufficiently separated frequency channels. Within the above framework this can be modeled by considering the graphG₁that models the adjacencies ofntransmitters, and takingG2=Kn, the complete graph onnvertices. The restrictions are clear: one asks

for a proper vertex coloring ofG2such that adjacent vertices inG1receive colors that differ by at least 2. We refer to [15,17]

for more particulars.

In [7], a situation is modeled in which the transmitters form a network in which a certain substructure of adjacent transmitters (called the backbone) is more crucial for the communication than the rest of the network. This means more restrictions are put on the assignment of frequency channels along the backbone than on the assignment of frequency channels to other adjacent transmitters.

Postponing the relevant definitions, we consider the problem of coloring the graphG2(that models the whole network)

with a proper vertex coloring such that the colors on adjacent vertices inG1(that models the backbone) differ by at least

λ

≥2. This is a continuation of the study in [7]. Throughout the paper we consider two types of backbones: matchings and disjoint unions of stars.

Matching backbones reflect the necessity to assign considerably different frequencies to pairwise very close (or most likely interfering) transmitters. This occurs in real world applications such as military scenarios, where soldiers or military vehicles carry two (or sometimes more) radios for reliable communication. Future applications include the use of sensors or sensor tags in clothes or on bodies.

For star backbones one could think of applications to sensor networks. If sensors have low battery capacities, the tasks of transmitting data are often assigned to specific sensors, called cluster heads, that represent pairwise disjoint clusters of sensors. Within the clusters there should be a considerable difference between the frequencies assigned to the cluster head and to the other sensors within the same cluster, whereas the differences between the frequencies assigned to the other sensors within the cluster, or between different clusters, are of secondary importance. This situation is well reflected by a backbone consisting of disjoint stars.

We refer the reader to [7,6] for a more extensive overview of related research, but we repeat the relevant definitions in the next section.

2. Terminology

For undefined terminology we refer to [3].

LetG=

(

V

,

E

)

be a graph, whereV=VGis a finite set of vertices andE=EGis a set of unordered pairs of two different

vertices, called edges. A functionf :V→ {1

,

2

,

3

, . . .

}is a vertex coloring ofVif|f

(

u

)

−f

(

v

)

| ≥1 holds for all edgesuv∈E. A vertex coloringf :V→ {1

, . . . ,

k}is called a k-coloring. We say thatf

(

u

)

is the color ofu. The chromatic number

χ(

G

)

is the smallest integerkfor which there exists ak-coloring. A setV0_⊆_V_{is independent if}_G_{does not contain edges with both end}

vertices inV0_{. By definition, a}_k_{-coloring partitions}_V_into_k_{independent sets}_V

1

, . . . ,

Vk.

LetHbe a spanning subgraph ofG, i.e.,H =

(

VG

,

EH

)

withEH ⊆ EG. Given an integer

λ

≥ 1, a vertex coloringf is a

λ

-backbone coloring of

(

G

,

H

)

, if|f

(

u

)

−f

(

v

)

| ≥

λ

holds for all edgesuv∈EH. A

λ

-backbone coloringf:V→ {1

, . . . , `

}is called

a

λ

-backbone

`

-coloring. The

λ

-backbone coloring number bbcλ(G

,

H

)

of

(

G

,

H

)

is the smallest integer

`

for which there exists a

λ

-backbone

`

-coloring. Since a 1-backbone coloring is equivalent to a vertex coloring, we assume from now on that

λ

≥2. Throughout the manuscript we will reserve the symbol “

`

” for

λ

-backbone

`

-colorings and the symbol “k” fork-colorings.

A path is a graphPwhose vertices can be ordered into a sequencev1

,

v2

, . . . ,

vnsuch thatEP= {v1v2

, . . . ,

vn−1vn}. A graph Gis called connected if for every pair of distinct verticesuandv, there exists a path connectinguandv. The length of a path is the number of its edges. If a graphGcontains a spanning subgraphHthat is a path, thenHis called a Hamiltonian path.

A cycle is a graphCwhose vertices can be ordered into a sequencev1

,

v2

, . . . ,

vnsuch thatEC= {v1v2

, . . . ,

vn−1vn

,

vnv1}. A tree is a connected graph that does not contain any cycles.

A complete graph is a graph with an edge between every pair of vertices. The complete graph onnvertices is denoted byKn. A graph is called bipartite if its vertices can be partitioned into two setsAandBsuch that each edge has one of its

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Fig. 1. Matching and star backbones.

endpoints incident with the setAand the other withB. A graphGis completep-partite if its vertices can be partitioned intop

nonempty independent setsV₁

, . . . ,

Vpsuch that its edge setEis formed by all edges that have one end vertex inViand the

other one inVjfor some 1≤i

<

j≤p.

Forq≥1, a starSqis a complete 2-partite graph with independent setsV1= {r}andV2with|V2| =q; the vertexris called

the root and the vertices inV₂are called the leaves ofSq. For the starS1we arbitrarily choose one of its two vertices to be the

root. In our context a matchingMis a collection of pairwise vertex-disjoint stars that are all copies ofS1. A matchingMof a

graphGis called perfect if it is a spanning subgraph ofG. We call a spanning subgraphHof a graphG

•a tree backbone ofGifHis a tree;

•a path backbone ofGifHis a Hamiltonian path;

•a star backbone ofGifHis a collection of pairwise vertex-disjoint stars;

•a matching backbone ofGifHis a perfect matching.

SeeFig. 1for an example of a graphGwith a matching backboneM(left) and a star backboneS(right). The thick edges are matching or star edges, respectively. The grey circles indicate root vertices of the stars inS.

Obviously, bbcλ(G

,

H

)

≥

χ(

G

)

holds for any backboneHof a graphG. In order to analyze the maximum difference between these two numbers the following values can be introduced.

Tλ(k

)

=max

bbcλ(G

,

T

)

|Tis a tree backbone ofG

,

and

χ(

G

)

=k

Pλ(k

)

=max

bbcλ(G

,

P

)

|Pis a path backbone ofG

,

and

χ(

G

)

=k

Sλ(k

)

=max

bbcλ(G

,

S

)

|Sis a star backbone ofG

,

and

χ(

G

)

=k

Mλ(k

)

=max

bbcλ(G

,

M

)

|Mis a matching backbone ofG

,

and

χ(

G

)

=k

.

3. Results

3.1. Existing results

The behavior ofTλ(k

)

andPλ(k

)

is determined in [7] as summarized in the following two results.

Theorem 1. T2

(

k

)

=2k−1 for allk≥1.

Theorem 2. The functionP2

(

k

)

takes the following values:

(a) for 1≤k≤4: _P₂

(

k

)

=2k−1; (b) P2

(

5

)

=8 andP2

(

6

)

=10;

(c) fork≥7 andk=4t: _P₂

(

4t

)

=6t;

(d) fork≥7 andk=4t+1: P2

(

4t+1

)

=6t+1;

(e) fork≥7 andk=4t+2: P2

(

4t+2

)

=6t+3;

(f) fork≥7 andk=4t+3: _P₂

(

4t+3

)

=6t+5.

The above theorems show the relation between the 2-backbone coloring number and the classical chromatic number in case the backbone is a tree or a path. We observe that in the worst case the 2-backbone coloring number roughly grows like 2kand 3k

/

2, respectively, where

χ

=k.

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3.2. Results of this paper

In this paper, we study the functionsSλ(k

)

andMλ(k

)

. By definition,Mλ(k

)

≤_Sλ(k

)

holds. We completely determine the behavior of these two functions. We first determine all valuesSλ(k

)

, and observe that they roughly grow like

(

2−_λ1

)

k. Then we determine all valuesMλ(k

)

and observe that they roughly grow like

(

2−_λ+2

1

)

k. Their precise behavior is summarized

in our two main theorems.

Theorem 4.For

λ

≥2 the functionSλ(k

)

takes the following values: (a)Sλ(2

)

=

λ

+1;

(b) for 3≤k≤2

λ

−3: Sλ(k

)

= d3k

2e +

λ

−2;

(c) for 2

λ

−1≤k≤2

λ

with

λ

=2: Sλ(k

)

=k+2

λ

−2; (d) for 2

λ

−2≤k≤2

λ

−1 with

λ

≥3: _Sλ(k

)

=k+2

λ

−2; (e) fork=2

λ

with

λ

≥3: Sλ(k

)

=2k−1;

(f) fork≥2

λ

+1: _Sλ(k

)

=2k− bk λc.

Theorem 5.For

λ

≥2 the functionMλ(k

)

takes the following values: (a) for 2≤k≤

λ

: Mλ(k

)

=k+

λ

−1;

(b) for

λ

+1≤k≤2

λ

: _M_λ

(

k

)

=2k−2; (c) fork=2

λ

+1: Mλ(k

)

=2k−3; (d) fork=t

(λ

+1

)

witht≥2: _M_λ

(

k

)

=2t

λ

;

(e) fork=t

(λ

+1

)

+cwitht≥2, 1≤c

<

λ+₂3 : Mλ(k

)

=2t

λ

+2c−1; (f) fork=t

(λ

+1

)

+cwitht≥2,λ+₂3≤c≤

λ

: Mλ(k

)

=2t

λ

+2c−2.

We note that there are many graphsGthat have a star backboneSsuch that bbcλ

(

G

,

S

) <

Sλ(χ(G

))

, or that have a matching backboneMsuch that bbcλ(G

,

M

) <

Mλ(χ(G

))

. As an example we mention the class of split graphs, e.g., graphs whose vertex set can be partitioned into a clique (i.e., a set of pairwise adjacent vertices) and an independent set, with possibly edges in between. In [8] we present (tight) upper bounds on the

λ

-star and

λ

-matching backbone coloring number for this graph class. These upper bounds are considerably smaller than the general bounds given inTheorems 4and5, respectively.

The rest of the paper is organized as follows. In the next section we consider the computational complexity of computing the

λ

-backbone coloring number for star and matching backbones. The fifth section gives the proof ofTheorem 4, and the sixth section gives the proof ofTheorem 5. There are many open problems about backbone colorings. We refer to [7] for details. In the last section of this paper we only focus on some open problems for matching backbone colorings for planar graphs.

4. Complexity results

The following decision problem can be defined.

λ

-Backbone Colorability (

`

) (

λ

-BBC (

`

)) Instance: A graphGwith a spanning subgraphH. Question: Is bbcλ(G

,

H

)

≤

`

?

Of course,

λ

-BBC (

`

) isNP-complete if

`

exceeds a certain value. In [7] it has been shown that the complexity of 2-BBC (

`

) restricted to instance graphsGwith a tree backboneHjumps from polynomially solvable toNP-complete between

`

=4 and

`

=5 (difficult even for path backbones). Here we restrict ourselves to instance graphsGwith a star backboneS. Star

λ

-Backbone Colorability (

`

) (

λ

-SBBC (

`

))

Instance: A graphGwith a star backboneS. Question: Is bbcλ(G

,

S

)

≤

`

?

Theorem 6.

λ

-SBBC (

`

) is polynomially solvable if

`

≤

λ

+1, and it isNP-complete if

`

≥

λ

+2 (even when restricted to matching backbones).

Proof. LetG =

(

V

,

E

)

be a graph with a star backboneS =

(

V

,

ES). For

`

≤

λ

no

λ

-backbone coloring exists. Now let

`

=

λ

+1. In any

λ

-backbone coloring with color set{1

,

2

, . . . , λ

+1}, colors 2

,

3

, . . . , λ

cannot be used at all, since each vertex is incident with an edge ofES. Hence bbcλ(G

,

S

)

=

λ

+1 if and only ifGis bipartite.

Let

`

≥

λ

+2. Obviously the problem

λ

-SBBC (

`

) is a member ofNP. We proveNP-completeness by reduction from Graphk-Colorability (cf. [14]): Given a graphG=

(

VG

,

EG

)

, does there exist ak-coloring ofG? This problem is known to be

NP-complete for any integerk≥3. We distinguish the following cases. Case 1

λ

+2≤

`

≤2

λ

−1.

Let

`

=

λ

+tfor some 2 ≤t≤

λ

−1, and letG=

(

VG,EG)be an instance of Graph 2t-Colorability. Letv1

,

v2

, . . . ,

vn

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that results from this is denoted byG0_{. The new edges form a matching backbone}_M_of_G0_{. We claim that}

_χ(

_G

₎

_≤ ₂_t_{if and}

only if bbcλ(G0

,

_M

)

_≤

`

_.

Assume that bbcλ(G0

,

_M

)

_≤

`

_{, and consider a}

λ

_-backbone

`

_-coloring_b_of_G0_{. Since all vertices in}_G0_{are incident with a}

matching edge, colorst+1

,

t+2

, . . . , λ

cannot be used at all. Then define a 2t-coloringcofGby:

•ifb

(

v

)

=jforj∈ {1

,

2

, . . . ,

t}:c

(

v

)

:=j;

•ifb

(

v

)

=

λ

+jforj∈ {1

,

2

, . . . ,

t}:c

(

v

)

:=t+j.

Next, assume that

χ(

G

)

≤ 2t, and consider a 2t-coloringf : VG → {1

, . . . ,

2t}. We define a

λ

-backbone

`

-coloring g:VG0→ {1

, . . . , `

}of

(

G0

,

M

)

by: •ifv∈VGandf

(

v

)

=jforj∈ {1

,

2

, . . . ,

t}:g

(

v

)

:=j; •ifv∈VGandf

(

v

)

=t+jforj∈ {1

,

2

, . . . ,

t}:g

(

v

)

:=

λ

+j; •ifg

(

vi)≤t:g

(

ui):=

`

; •Ifg

(

vi)≥

λ

+1:g

(

ui):=1. Case 2

`

≥2

λ

.

LetG=

(

VG

,

EG

)

be an instance of Graph

`

-Colorability, and denote the vertices inVGbyv1

,

v2

, . . . ,

vn. We creatennew

verticesu1

,

u2

, . . . ,

unand introduce new edgesviui(i=1

,

2

, . . . ,

n). The graph that results from this is denoted byG0. The

new edges form a matching backboneMofG0_{. We complete the proof by showing that}

χ(

_G

)

_≤

`

if and only if bbcλ

(

G0

,

_M

)

_≤

`

_.

Indeed, assume that bbcλ(G0

,

M

)

≤

`

and consider such a

λ

-backbone

`

-coloring. Then the restriction to the vertices in

VGyields an

`

-coloring ofG. Next assume that

χ(

G

)

≤

`

, and consider an

`

-coloringf :VG → {1

, . . . , `

}. We extendf to a

λ

-backbone

`

-coloring of

(

G0

,

_M

)

_{: If}_f

(

_vi)_≤

λ

_{, then vertex}_u

iis colored with color

`

, and otherwise it is assigned color 1. This

completes the proof.

5. Proof ofTheorem 4

We proveTheorem 4in two steps. First we show that bbcλ

(

G

,

S

)

for any graphGwith arbitrary star backboneSis at most the value ofSλ

(χ(

G

))

as given inTheorem 4. Next we present a class of graphsGthat have a star backboneSsuch that bbcλ(G

,

S

)

is at least the value ofSλ(χ(G

))

that is given inTheorem 4. This way we obtain coinciding upper and lower bounds onSλ(k

)

that prove the theorem.

5.1. Proof of the upper bounds

LetG=

(

V

,

E

)

be a graph with

χ(

G

)

=kand letV1

, . . . ,

Vkdenote the corresponding independent sets in ak-coloring. Let S=

(

V

,

ES

)

be a star backbone ofG. Ifk=2 thenGis bipartite, and we use colors 1 and

λ

+1. This proves the upper bound

for case (a) of the theorem. Case (b) 3≤k≤2

λ

−3.

Consider the following color sets:

•Ci= {i

,

k+

λ

−1−i}fori=1

, . . . ,

bk₂c; •Ci= {i

,

2k+

λ

−1−i}fori= bk₂c +1

, . . . ,

k.

The union of these k color sets consists of 2k colors, namely the colors in {1

, . . . ,

k} together with the colors in

{k+

λ

−1− bk

2c

, . . . ,

2k+

λ

−1−

(

b k

2c +1

)

}. The largest color used is 2k+

λ

−1−

(

b k

2c +1

)

= d 3k

2e +

λ

−2.

We construct a

λ

-backbone coloring of

(

G

,

S

)

such that every vertex inVi

(

i=1

, . . . ,

k

)

is colored with a color inCi. Since

the vertex subsetsViare independent, we will obtain a vertex coloring this way. To show that we can obtain a

λ

-backbone

(

d3k

2e +

λ

−2

)

-coloring this way, we have to be a bit more careful.

For 1≤i≤ bk₂c

,

a root vertex inViis colored with the first color ofCi. Forbk₂c +1≤i≤k

,

a root vertex inViis colored

with the second color ofCi.

The leaves in a setVjof a star with a root in a setVifor 1≤i≤ bk₂care colored with the second color ofCj. This does not

give any conflict, since the smallest gap appears if the root vertex is inVbk

2cand one of its leaves is inVbk2c−1, or the other way around. In both cases this gap isk+

λ

−1− bk

2c −

(

b k

2c −1

)

=k+

λ

−2b k 2c ≥

λ

.

The leaves in a setVjof a star with a root in a setViforb₂kc +1≤i≤kare colored with the first color ofCj. This is possible,

since the smallest gap appears if the root vertex is inVkand one of its leaves is inVk−1, or the other way around. In both cases

this gap is 2k+

λ

−1−k−

(

k−1

)

=

λ

. Hence, we indeed have obtained a desired

λ

-backbone

(

d3k

2e +

λ

−2

)

-coloring of

(

G

,

S

)

.

Case (c)

λ

=2

,

k=3 or

λ

=2

,

k=4.

For proving thatS2

(

3

)

≤5 we use color setsC1= {1},C2= {3},C3= {5}. We color the vertices ofV1by 1, the vertices of V2by 3, and the vertices ofV3by 5. This gives us a 2-backbone 5-coloring of

(

G

,

S

)

.

For proving thatS2

(

4

)

≤6 we use color setsC1= {1},C2= {2

,

3},C3= {4

,

5}andC4= {6}. We use color 1 for all vertices

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star with root inV1, we colorvby 3. Otherwise we colorvby 2. Ifw∈V3is a leaf of a star with root inV4, we colorwby 4.

Otherwise we colorwby 5. This gives us a 2-backbone 6-coloring of

(

G

,

S

)

. For the cases (d)–(f) we need the following lemma.

Lemma 7. LetG=

(

V

,

E

)

be a graph with

χ(

G

)

=kand letV1

, . . . ,

Vkdenote the corresponding independent sets in ak-coloring. LetS =

(

V

,

ES)be a star backbone of G. Fori = 1

, . . . ,

p, letCi = {ai}be a set consisting of one color. Forj = 1

, . . . ,

q, let Dj = {bj

,

cj}be a set consisting of two colors. Letd = max{a1

, . . . ,

ap

,

b1

, . . . ,

bq

,

c1

, . . . ,

cq}. Then

(

G

,

S

)

has a

λ

-backbone d-coloring if the following conditions are satisfied:

(i) p+q=k.

(ii) Ci∩Cj= ∅for all 1≤i

<

j≤p.

(iii) Di∩Dj= ∅for all 1≤i

<

j≤q.

(iv) Ci∩Dj= ∅for all 1≤i≤pand 1≤j≤q.

(v) |ai−aj| ≥

λ

for all 1≤i

<

j≤p.

(vi) |ai−bj| ≥

λ

for all 1≤i≤pand 1≤j≤q.

(vii)|bj−cj| ≥2

λ

−1 for all 1≤j≤q.

Proof. Due to (i), we can map each vertex inVito a color inCifori = 1

, . . . ,

pand each vertexvinVjto a color inDj−p

forj=p+1

, . . . ,

k. Since the setsViare independent, conditions (ii)–(iv) imply that this way we are guaranteed to obtain

a vertex coloring ofGwith colors in{1

, . . . ,

d}. Below we explain how we can refine this strategy such that we obtain a

λ

-backboned-coloringfof

(

G

,

S

)

.

So far only the colors of vertices inVifori=1

, . . . ,

phave been fixed by a coloringfas above. Due to (v),|f

(

u

)

−f

(

v

)

| ≥

λ

for all star edgesuvwithu∈Vi,v∈Vjfor some 1≤i

<

j≤p.

We letfcolor a root vertex inVjforp+1≤j≤kwith colorbj−p. Due to (vi), we find that|f

(

v

)

−f

(

u

)

| ≥

λ

holds for all

star edgesuvwith leafuinVifor some 1≤i≤pand rootvinVjfor somep+1≤j≤k.

What about the other vertices? They are all leaf vertices in setsVjwithp+1≤j≤k. Letv∈Vjwithp+1≤j≤kbe a

leaf vertex of a starSwith rootw. Letxbe the color assigned tow. Then colorsx−

λ

+1

, . . . ,

x+

λ

−1 are forbidden colors forv. The distance betweenx+

λ

−1 andx−

λ

+1 is 2

λ

−2. Since the two colors inDj−phave pairwise distance at least

2

λ

−1 due to (vii), at least one of them is feasible forv. This finishes the proof of the lemma. Case (d) 2

λ

−2≤k≤2

λ

−1 with

λ

≥3.

We use color sets:

• C₁= {k};

• Dj= {j

,

j+2

λ

−1}forj=1

, . . . ,

k−1.

We will show that these color sets satisfy the conditions ofLemma 7. First note that thesekcolor sets are pairwise disjoint: the union of these sets consists of all the colors in{1

, . . . ,

k}together with all the colors in{2

λ, . . . ,

k+2

λ

−2}. We setbj:=jfor 1≤j≤ dk₂e −1. Thena1−bj=k−j≥k− d₂ke +1≥

λ

. for 1≤j≤ d₂ke −1. We setbj:=j+2

λ

−1 for dk

2e ≤j≤k−1. Thenbj−a1=j+2

λ

−1−k≥ dk₂e +2

λ

−1−k≥

λ

fordk₂e ≤j≤k−1. We observe that the two colors bj,cjin any setDjhave pairwise distance 2

λ

−1. Hence, all conditions ofLemma 7are satisfied. This implies that

(

G

,

S

)

has

a

λ

-backbone

(

k+2

λ

−2

)

-coloring. Case (e)k=2

λ

with

λ

≥3.

We use color sets:

• C1= {4

λ

−1};

• Dj= {j

,

2

λ

−1+j}forj=1

, . . . ,

k−1.

We will show that these color sets satisfy the conditions ofLemma 7. Note that thesekcolor sets are pairwise disjoint: the union of these sets consists of all the colors in{1

, . . . ,

k−1}together with all the colors in{2

λ, . . . ,

4

λ

−1}. We set

bj := jfor 1 ≤j ≤k−1. Thena1−bj = 4

λ

−1−j ≥4

λ

−1−

(

k−1

)

= 2

λ

≥

λ

for 1 ≤j ≤k−1. We note that the

difference between the two colorsbjandcjin any setDjis equal to 2

λ

−1+j−j=2

λ

−1. Hence, all conditions ofLemma 7

are satisfied. This implies that

(

G

,

S

)

has a

λ

-backbone

(

4

λ

−1

)

-coloring. Case (f)k≥2

λ

+1.

We use color sets:

• Ci= {

(

i−1

)λ

+1}fori=1

, . . . ,

b_λkc; • Dj= {d_λ−jλ₁e

,

j+k}forj=1

, . . . ,

b_λkc

(λ

−1

)

;

(7)

Fig. 2. The graph T(9,3)with star backbone S.

We will show that thesekcolor sets satisfy the conditions ofLemma 7. Ifj=s

(λ

−1

)

+tfor some integerss≥0 and 0≤t≤

λ

−2, thend jλ

λ−1eis equal tos

λ

in caset=0 and tos

λ

+t+1 in caset

>

0. ThenCi∩Djis empty for all 1≤i≤ bk_λcand

1≤j≤ bk_λc

(λ

−1

)

. Hence thekcolor sets as defined above are pairwise disjoint, and cover the whole range 1

, . . . ,

2k− bk_λc. We observe that two colorsai

,

ajin two different setsCiandCjare at least

λ

apart from each other. We definebj:=j+k

for 1≤j≤k− bk

λc. The smallest gap between a colorbjand a coloraiis 1+k−

((

bk_λc −1

)λ

+1

)

=k− bk_λc

λ

+

λ

≥

λ

.

Forj=1

, . . . ,

b_λkc

(λ

−1

)

, the distance between two colors in a color setDjis j+k− _j

λ

−1 =j+k− j+ j

λ

−1 =k− _j

λ

−1 ≥k− & bk_λc

(λ

−1

)

λ

−1 ' =k− _k

λ

.

Also the distance between two colors in a color setDjforj= b_λkc

(λ

−1

)

+1

, . . . ,

k− b_λkcis at leastk− b_λkc. We deduce that k− _k

λ

= _k

(λ

₋₁

)

λ

≥

(

₂

λ

₊₁

)(λ

₋₁

)

λ

= 2

λ

−1−1

λ

=2

λ

−1

.

Hence, all conditions ofLemma 7are satisfied. This implies that

(

G

,

S

)

has a

λ

-backbone

(

2k− bk_λc

)

-coloring. 5.2. Proof of the lower bounds

Let

λ

≥2. The casek=2 is trivial. Fork≥3, we consider the Turán graphT

(

k2

_,

_k

₎

_{, i.e., a complete}_k_{-partite graph that}

consists ofkindependent setsV1

, . . . ,

Vkthat are all of cardinalityk. LetS=

(

V

,

ES)be a star backbone ofT

(

k2

,

k

)

that consists

ofkstarsSk−1. EachVicontains exactly one root vertex of some star inSand its otherk−1 vertices are leaves of stars rooted

ink−1 different setsVj6=Vi. SeeFig. 2for an example of the graphT

(

9

,

3

)

with star backboneS; the setsViare indicated

and the thick edges are the star edges. For our case analysis we first prove a number of results for an arbitrary

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

.

Letfbe a

λ

-backbone

`

-coloring of

(

T

(

k2

_,

_k

_),

_S

₎

_{. Since}_T

₍

_k2

_,

_k

₎

_{is complete}_k_{-partite, any color that shows up in some set}_V i

cannot show up in anyVjwithj6=i. We denote byCithe set of colors that are used on vertices inVi. If|Ci| =1, thenViis called monochromatic, and if|Ci| ≥2, thenViis called polychromatic. We denote bys1ands2the number of monochromatic and

polychromatic sets, respectively. Then we immediately haves₁+s₂=kands₁+2s₂≤

`

implying the following observation.

Observation 8. Letf be a

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

withs1monochromatic sets. Thens1≥2k−

`

holds.

Since all stars inShave (exactly) one leaf in any set that does not contain their root vertex, we immediately have the following.

Observation 9. Letf be a

λ

-backbone

`

-coloring of

(

T

(

k2

_,

_k

_),

_S

₎

_{. Let}_x_{be the color for the root in set}_V

i. LetVj

(

j 6= i

)

be a monochromatic set colored byy. Then the distance betweenxandyis at least

λ

.

We useObservation 9to prove the following lemma.

Lemma 10. Letfbe a

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

withs1monochromatic sets ands2polychromatic sets. Then

`

≥

(

k−1

)λ

+1 ifs2=0;

s1

(λ

−1

)

+k ifs2

>

0

.

(1)

Proof. Supposes₂=0. Thens₁=k, and byObservation 9there are at least

(

k−1

)

gaps of at least

λ

−1 colors that cannot be used to color thekroots. Then the total number of colors needed is at least

(

k−1

)(λ

−1

)

+k=

(

k−1

)λ

+1.

Ifs₂

>

0,Observation 9implies that there are at leasts₁gaps of at least

λ

−1 colors. In this case the total range of colors is at leasts1

(λ

−1

)

+k.

A root in a monochromatic set is called monochromatic as well. A root color is a color that is used for a root. Recall that all stars inShave (exactly) one leaf in any set that does not contain their root vertex. Then we can easily make the following observation.

(8)

Observation 11.Letfbe a

λ

-backbone

`

-coloring of

(

T

(

k2

,

_k

),

_S

)

_{. Let}_x_{be the color for the root in}_V

i. Then there are (at least) k−1 different colorsy1

, . . . ,

yi−1

,

yi+1

, . . . ,

ykthat have distance at least

λ

tox: everyVj

(

j6=i

)

contains a vertex with coloryj.

Due toObservation 11we can prove the following lemma.

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

. If

`

≤k+2

λ

−3, then only colors fromA= {1

, . . . , `

−k−

λ

+2} andB= {k+

λ

−1

, . . . , `

}can be assigned to root vertices.

Proof. Suppose a rootvis assigned colorcwithcin{

`

−k−

λ

+3

, . . . ,

k+

λ

−2}. ByObservation 11there have to be at leastk−1 colors with distance at least

λ

fromc. If

λ

+1≤c ≤

`

−

λ

, only colors in{1

, . . . ,

c−

λ

}and in{c+

λ, . . . , `

}

can be used. These sets together containc−

λ

+

`

−

(

c+

λ)

+1 =

`

−2

λ

+1 ≤ k−2 colors. Hence eitherc ≤

λ

or

c ≥

`

−

λ

+1 holds. Ifc ≤

λ

, then only colors in{c+

λ, . . . , `

}are at distance at least

λ

. The cardinality of this set is

`

−

(

c+

λ)

+1≤

`

−

(`

−k−

λ

+3

)

−

λ

+1=k−2. Ifc≥

`

−

λ

+1, then only colors in{1

, . . . ,

c−

λ

. The cardinality of this set isc−

λ

≤k+

λ

−2−

λ

=k−2.

We are now ready to make our case analysis. Case (b) 3≤k≤2

λ

−3.

Suppose there exists a

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

with

`

= d3k

2e +

λ

−3 colors. Then

`

= d 3k

2e +

λ

−3 ≤ k+dk

2e+

λ

−3≤k+2

λ

−3 and byLemma 12only colors inA= {1

, . . . ,

d k

2e−1}and colors inB= {k+

λ

−1

, . . . ,

d 3k

2e+

λ

−3}

can be used on roots. Each root is in a different independent setVi. Therefore the number of different root colors is equal to k. However, the total number of colors inAunited withBis 2

(

dk

2e −1

) <

k. This contradiction shows that we must have

`

≥ d3k

2e +

λ

−2.

Case (c, d) 2

λ

−1≤k≤2

λ

with

λ

=2 or 2

λ

−2≤k≤2

λ

−1 with

λ

≥3.

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

with

`

= k+2

λ

−3 colors. ByLemma 12, only colors in

A= {1

, . . . , λ

−1}andB= {k+

λ

−1

, . . . ,

k+2

λ

−3}may be used on roots. ByObservation 8,s1≥2k−

`

=k−2

λ

+3≥

2

λ

−2−2

λ

+3≥1. So there exists at least one monochromatic set. Letybe the (root) color used on this set. Without loss of generality we may assume thatyis inA. ByObservation 9, all otherk−1 root colors must be inB. However,Bcontains

λ

−1

<

k−1 colors. This contradiction shows that we must have

`

≥k+2

λ

−2. Case (e)k=2

λ

with

λ

≥3.

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

with 2k−2=4

λ

−2 colors. Ifs2=0, then by(1)we would

have 2k−2=

`

≥

(

k−1

)λ

+1≥3

(

k−1

)

+1=3k−2. Hences₂

>

0. ByObservation 8,s1≥2k−

`

=2. Together with(1)we then deduce that

4

λ

−2=

`

≥s1

(λ

−1

)

+k≥2

(λ

−1

)

+2

λ

=4

λ

−2

.

Hence we find thats1 = 2, and

`

= s1

(λ

−1

)

+k. Due toObservation 9, there are only three feasible ways to choosek

different root colors:

1. monochromatic roots: 1

, λ

+1, other roots: 2

λ

+1

, . . . ,

4

λ

−2;

,

4

λ

−2, other roots:

λ

+1

, . . . ,

3

λ

−2;

λ

−2

,

4

λ

−2, other roots: 1

, . . . ,

2

λ

−2.

Consider situation 1. Since color 2

λ

+1 is a root color, byObservation 11, in every other color set there must be at least one color that has distance at least

λ

to color 2

λ

+1. This necessary condition is already met for the sets with root color 1, root color

λ

+1 or root colors 3

λ

+1

, . . . ,

4

λ

−2. However, the sets with root colors 2

λ

+2

, . . . ,

3

λ

need an extra color. Hence, we need

λ

−1 extra colors that have distance at least

λ

to color 2

λ

+1. There are exactly

λ

−1 such colors available, namely colors 2

, . . . , λ

. So one of the colors 2

, . . . , λ

must be in the same set with color 2

λ

+2.

Simultaneously, since color 2

λ

+2 is also a root color, in every other color set there must be at least one color that has distance at least

λ

to color 2

λ

+2. This condition is not met yet for the sets with root color 2

λ

+1 or root colors 2

λ

+3

, . . . ,

3

λ

+1. To satisfy the condition, we need

λ

extra colors that have distance at least

λ

to color 2

λ

+2. The only available colors are colors 2

, . . . , λ

and color

λ

+2. This implies that none of the colors 2

, . . . , λ

can be in the same set with color 2

λ

+2. This contradiction shows that we must have

`

≥2k−1.

Consider situation 2. Since color

λ

+1 is a root color, byObservation 11, in every other color set there must be at least one color that has distance at least

λ

to color

λ

+1. This necessary condition is already met for the sets with root color 1, root color 4

λ

−2 or root colors 2

λ

+1

, . . . ,

3

λ

−2. However, the sets with root colors

λ

+2

, . . . ,

2

λ

need an extra color. Hence, we need

λ

−1 extra colors that have distance at least

λ

to color

λ

+1. There are exactly

λ

−1 such colors available, namely colors 3

λ

−1

, . . . ,

4

λ

−3. So one of the colors 3

λ

−1

, . . . ,

4

λ

−3 must be in the same set with color

λ

+2.

Simultaneously, since color

λ

+2 is also a root color, in every other color set there must be at least one color that has distance at least

λ

to color

λ

+2. This condition is not met yet for the sets with root color

λ

+1 or root colors

λ

+3

, . . . ,

2

λ

+1. To satisfy the condition, we need

λ

extra colors that have distance at least

λ

to color

λ

+2. The only available colors are color 2 and the colors 3

λ

−1

, . . . ,

4

λ

−3. This implies that none of the colors 3

λ

−1

, . . . ,

4

λ

−3 can be in the same set with color

(9)

By symmetry, situation 3 yields the same conclusion as situation 1. Hence, we conclude that any

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

has

`

≥2k−1.

Case (f)k≥2

λ

+1.

λ

-backbone

`

-coloring of

(

T

(

k2

,

k

),

S

)

with

`

=2k− bk_λc −1 colors. Supposes2=0. Then there

are only monochromatic sets, i.e.,s₁ = k. By(1)the total number of colors needed is at least

(

k−1

)λ

+1. However, the difference between this number and

`

is

(

k−1

)λ

+1− 2k− _k

λ

−1 =k

(λ

−2

)

+ _k

λ

−

λ

+2≥2

λ

2−4

λ

+2

>

0

.

Hences2

>

0. Writek = a

λ

+rfor some integersa≥ 2 and 0≤ r ≤

λ

−1. ByObservation 8,s1 ≥ 2k−

`

= b_λkc +1

holds. Together with(1)this implies that we need at least

(

b_λkc +1

)(λ

−1

)

+kcolors. However, the difference between this number and

`

is

(

b_λkc +1

)(λ

−1

)

+k−

(

2k− bk_λc −1

)

= b_λkc

λ

+

λ

−k=

λ

−r

>

0. This contradiction shows that we must have

`

≥2k− b_λkc.

This finishes the proof of the lower bounds, and we have completed the proof ofTheorem 4.

6. Proof ofTheorem 5

We proveTheorem 5in two steps. First we show that bbcλ(G

,

M

)

for any graphGwith arbitrary matching backboneMis at most the value ofMλ(χ(G

))

as given inTheorem 5. Next we present a class of graphsGthat have a matching backboneM

such that bbcλ

(

G

,

M

)

is at least the value ofMλ(χ(G

))

that is given inTheorem 5. This way we obtain coinciding upper and lower bounds onMλ(k

)

proving the theorem.

6.1. Proof of the upper bounds

LetG=

(

V

,

E

)

be a graph with

χ(

G

)

=kand letV₁

, . . . ,

Vkdenote the corresponding independent sets in ak-coloring. Let M=

(

V

,

EM)be a matching backbone ofG.

Case (a) 2≤k≤

λ

.

Ifk=2 thenGis bipartite, and we use colors 1 and

λ

+1. Letk≥3. Letuvbe a matching edge withu∈Viandv∈ Vj

for some 1≤i

<

j≤k. We colorubyiandvby

λ

+j−1. Then the difference between the colors ofuandvis at least

λ

. So vertices inV1get color 1, vertices inViwith 2≤i≤k−1 get colorior

λ

+i−1, and vertices inVkget color

λ

+k−1. Hence,

we have obtained a

λ

-backbone

(λ

+k−1

)

-coloring of

(

G

,

M

)

. Case (b)

λ

+1≤k≤2

λ

.

Letuvbe a matching edge withu∈Viandv∈Vjfor some 1≤i

<

j≤k. We colorubyiandvbyk+j−2. This way we

obtain a

λ

-backbone

(

2k−2

)

-coloring of

(

G

,

M

)

.

For the cases (c)–(f) we need the following lemma. Observe that this lemma is exactly the same asLemma 7except that condition (vi) ofLemma 7could be dropped. The first two paragraphs of the proof can be copied from the proof ofLemma 7.

Lemma 13. LetG=

(

V

,

E

)

be a graph with

χ(

G

)

=kand letV1

, . . . ,

Vkdenote the corresponding independent sets in ak-coloring. LetM=

(

V

,

EM

)

be a matching backbone ofG. Fori= 1

, . . . ,

p, letCi = {ai}be a set consisting of one color. Forj= 1

, . . . ,

q, letDj = {bj

,

cj}be a set consisting of two colors. Letd=max{a1

, . . . ,

ap

,

b1

, . . . ,

bq

,

c1

, . . . ,

cq}. Then

(

G

,

M

)

has a

λ

-backbone d-coloring if the following conditions are satisfied:

(i) p+q=k.

(ii) Ci∩Cj= ∅for all 1≤i

<

j≤p.

(iii) Di∩Dj= ∅for all 1≤i

<

j≤q.

(iv) Ci∩Dj= ∅for all 1≤i≤pand 1≤j≤q.

(v) |ai−aj| ≥

λ

for all 1≤i

<

j≤p.

(vi) |bj−cj| ≥2

λ

−1 for all 1≤j≤q.

Proof. Letuvbe a matching edge, whereu∈Viwith 1≤i≤p, andv∈Vjwithp+1≤j≤k. Thenuhas colorai. Then colors ai−

λ

+1

, . . . ,

ai+

λ

−1 are forbidden colors forv. The distance betweenai+

λ

−1 andai−

λ

+1 is 2

λ

−2. Since the two

colors inDj−phave pairwise distance at least 2

λ

−1 due to (vi), at least one of them is feasible forv.

For all matching edgesuvwithu∈Viandv∈Vjfor somep+1≤i

<

j≤kwe chooseuto be the root. We coloruwith bi. The remaining vertices, whose colors have not yet been fixed, are all leaf vertices in setsVjwithp+1≤j≤k. Again due

to (vi) we can color them with a feasible color fromDj−p. This finishes the proof of the lemma.

Below we show which color sets we use for each case. To check that these color sets satisfy the conditions ofLemma 13 is a simple exercise and left to the reader.

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Fig. 3. The graph T(6,2)with matching backbone M. Case (c)k=2

λ

+1.

We use color sets:

• Ci= {i

λ

+1}fori=0

, . . . ,

3; • D₁_,j= {j

,

2

λ

+j}forj=2

, . . . , λ

;

• D2,j= {

λ

+j

,

3

λ

+j}forj=2

, . . . , λ

−1 and

λ

≥3. Case (d)k=t

(λ

+1

)

witht≥2.

We use color sets:

• Ci= {i

λ

+1}fori=0

, . . . ,

2t−1;

• Di,j= {i

λ

+j

, (

t+i

)λ

+j}fori=0

, . . . ,

t−1 andj=2

, . . . , λ

. Case (e)k=t

(λ

+1

)

+cwitht≥2, 1≤c

<

λ+₂3.

We use color sets:

• Ci= {i

λ

+1}fori=0

, . . . ,

2t; • D0,j= {j

,

2t

λ

+2j−2}forj=2

, . . . ,

candc≥2; • D₀_,j= {j

,

t

λ

+j}forj=c+1

, . . . , λ

andc

< λ

; • Di,j= {i

λ

+j

, (

t+i

)λ

+j}fori=1

, . . . ,

t−1 andj=2

, . . . , λ

; • D_t,j= {t

λ

+j

,

2t

λ

+2j−1}forj=2

, . . . ,

candc≥2. Case (f)k=t

(λ

+1

)

+cwitht≥2,λ+3 2 ≤c≤

λ

.

We use color sets:

• Ci= {i

λ

+1}fori=0

, . . . ,

2t; • C2t+1= {2t

λ

+2c−2}; • D₀_,j= {j

,

2t

λ

+2j−2}forj=2

, . . . ,

c−1; • D0,j= {j

,

t

λ

+j}forj=c

, . . . , λ

; • Di,j= {i

λ

+j

, (

t+i

)λ

+j}fori=1

, . . . ,

t−1 andj=2

, . . . , λ

; • Dt,j= {t

λ

+j

,

2t

λ

+2j−1}forj=2

, . . . ,

c−1.

6.2. Proof of the lower bounds

Let

λ

≥ 2. Fork ≥ 2, we consider the Turán graphT

(

k2₋_k

_,

_k₋₁

₎

_{, i.e., a complete}_k_{-partite graph that consists of}_k

independent setsV1

, . . . ,

Vkthat are all of cardinalityk−1. For 1≤i≤k, let{vi,j |1≤j≤k

,

j6= i}be the vertices ofVi,

and letMbe a matching backbone ofT

(

k2−k

,

k−1

)

such thatEM= {vi,jvj,i|1≤i

<

j≤k}. SeeFig. 3for an example of the

graphT

(

6

,

2

)

with matching backboneM. SoVT(6,2)= {v1,2

,

v1,3

,

v2,1

,

v2,3

,

v3,1

,

v3,2}andEM= {v1,2v2,1

,

v1,3v3,1

,

v2,3v3,2}. For

our case analysis we first prove a number of results for an arbitrary

λ

-backbone

`

-coloring of

(

T

(

k2−k

,

k−1

),

M

)

. Consider some

λ

-backbone

`

-coloringf of

(

T

(

k2−k

,

k−1

),

M

)

. SinceT

(

k2−k

,

k−1

)

is completek-partite, any color that shows up in some setVicannot show up in anyVjwithj6= i. As in the star backbone case, we denote byCithe set of

colors that are used on vertices inVi. Recall that a setViis called monochromatic if|Ci| =1, and polychromatic if|Ci| ≥2.

Again we denote bys1ands2the number of monochromatic and polychromatic sets, respectively. Letm≤

`

be the number

of different colors used onV. Then we immediately haves1+s2=kands1+2s2≤mimplying the following observation.

λ

-backbone

`

-coloring of

(

T

(

k2₋_k

_,

_k₋₁

_),

_M

₎

_using_m_{colors and with}_s

1monochromatic sets. Then s1≥2k−mholds.

Since there exists a matching edge between any two independent setsViandVj, we obtain the following observation.

λ

-backbone

`

-coloring of

(

T

(

k2−k

,

k−1

),

M

)

. If colorxis assigned to a monochromatic setVi, and coloryis assigned to a monochromatic setVj, then the distance betweenxandyis at least

λ

.

(11)

λ

-backbone

`

-coloring of

(

T

(

k2₋_k

,

_k₋₁

),

_M

)

_{. Then}

`

≥ 2

λ

k

λ

+1−

λ

−1

λ

+1

.

(2)

Proof. Letmbe the number of different colors thatf uses.Observation 15yields

`

≥

λ(

s1 −1

)

+1. Together with Observation 14andm≤

`

, we obtain

`

≥

λ(

s1−1

)

+1≥

λ(

2k−m−1

)

+1≥

λ(

2k−

`

−1

)

+1, which is equivalent to

inequality(2).

Also the following lemma is useful.

Lemma 17. Let f be a

λ

-backbone

`

-coloring of

(

T

(

k2− k

,

k−1

),

M

)

. If

`

≤ k+2

λ

− 3 then only colors fromA = {1

, . . . , `

−k−

λ

+2}andB= {k+

λ

−1

, . . . , `

}can be assigned to monochromatic sets.

Proof. Suppose a vertexvfrom a monochromatic set is assigned colorcwithcin{

`

−k−

λ

+3

, . . . ,

k+

λ

−2}. Recall that there exists a matching edge between any two independent sets Vi and Vj. Then there are at least k− 1 colors

that have distance at least

λ

toc. If

λ

+1 ≤ c ≤

`

−

λ

, only colors in{1

, . . . ,

c−

λ

}and in{c+

λ, . . . , `

}can be used. These sets together containc−

λ

+

`

−

(

c+

λ)

+1 =

`

−2

λ

+ 1 ≤ k− 2 colors. Hence eitherc ≤

λ

or

c ≥

`

−

λ

+1 holds. Ifc ≤

λ

, then only colors in{c+

λ, . . . , `

λ

. The cardinality of this set is

`

−

(

c+

λ)

+1≤

`

−

(`

−k−

λ

+3

)

−

λ

+1=k−2. Ifc≥

`

−

λ

+1, then only colors in{1

, . . . ,

c−

λ

. The cardinality of this set isc−

λ

≤k+

λ

−2−

λ

=k−2.

We are now ready to make our case analysis. Case (a) 2≤k≤

λ

.

The casek=2 is trivial. Letk≥3. Suppose

(

T

(

k2−k

,

k−1

),

M

)

has a

λ

-backbone

`

-coloring with

`

=k+

λ

−2 colors. ByLemma 17, we find thats1=0. Colorsk−1

, . . . , λ

cannot be used at all, since there is no color in{1

, . . . , λ

+k−2}that

has distance at least

λ

to one of them. So we can only use colors in{1

, . . . ,

k−2}and{

λ

+1

, . . . , λ

+k−2}. Then the total numbermof different colors is at most 2

(

k−2

)

. Hence, byObservation 14we find thats1≥2k−m

>

0. This contradiction

shows that we must have

`

≥k+

λ

−1. Case (b)

λ

+1≤k≤2

λ

.

Suppose

(

T

(

k2₋_k

,

_k₋₁

),

_M

)

_{has a}

λ

_-backbone

`

_{-coloring with}

`

₌ ₂_k₋_{3 colors. By}_{Observation 14}_{, we find that} s1 ≥ 2k−m ≥ 2k−

`

≥ 3 must hold. ByLemma 17, only monochromatic colors in A = {1

, . . . ,

k−

λ

−1}and B = {k+

λ

−1

, . . . ,

2k−3}can be used. Both sets havek−

λ

−1 ≤

λ

−1 elements. Then, byObservation 15, at most one color inAand at most one color inBcan be used for monochromatic sets. Hence we find thats1≤2. This contradiction

shows that

`

≥2k−2. Case (c)k=2

λ

+1.

Analogously to the proof of the previous case we can show that

`

≥2k−3 must hold for any

λ

-backbone

`

-coloring of

(

T

(

k2₋_k

_,

_k₋₁

_),

_M

₎

_.

Case (d)k=t

(λ

+1

)

witht≥2. Inequality(2)yields

`

≥2t

λ

−λ−_λ+1

1=2t

λ

−1+ 2

λ+1for any

λ

-backbone

`

-coloring of

(

T

(

k

2₋_k

_,

_k₋₁

_),

_M

₎

_{. Since}

_`

_{is an}

integer, this implies that

`

≥2t

λ

.

Case (e)k=t

(λ

+1

)

+cwitht≥2 and 1≤c

<

λ+₂3.

Inequality(2)yields

`

≥2t

λ

+_λ+2λc₁−_λ+λ−₁1=2t

λ

+2c−1+2_λ+−2₁c

>

2t

λ

+2c−1+2−_λ+λ−₁3=2t

λ

+2c−2 for any

λ

-backbone

`

-coloring of

(

T

(

k2−k

,

k−1

),

M

)

. Since

`

is an integer, we have found that

`

≥2t

λ

+2c−1. Case (f)k=t

(λ

+1

)

+cwitht≥2 andλ+₂3 ≤c≤

λ

.

Inequality(2)yields

`

≥2t

λ

+_λ+2λc 1−λ− 1 λ+1 =2t

λ

+2c−1+ 2−2c λ+1 ≥2t

λ

+2c−1+ 2−2λ λ+1 =2t

λ

+2c−3+ 2 λ+1for any

λ

-backbone

`

-coloring of

(

T

(

k2−k

,

k−1

),

M

)

. Since

`

is an integer, this implies that

`

≥2t

λ

+2c−2.

This finishes the proof of the lower bounds, and we have completed the proof ofTheorem 5.

7. Matching backbones for planar graphs

7.1. Implications of the four color theorem

In the last section of this paper we focus on some open problems for matching backbone colorings on planar graphs. For simplicity we assume

λ

=2. The Four Color Theorem together withTheorem 5implies that bbc2

(

G

,

M

)

≤6 holds for any

planar graphGwith a matching backboneM. It seems likely that this bound 6 is not the best possible. However, the planar graphG₁with indicated matching backboneMconsisting of edgesab0

_,

_bc0

_,

_cd0

_,

_da0

as inFig. 4shows that one cannot improve this bound to 4.

We prove here that we cannot find a backbone coloring of

(

G1

,

M

)

with color set{1

,

2

,

3

,

4}. First of all observe thatG1

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Fig. 4. A graph G1with a matching backbone M such that bbc2(G1,M)=5.

adding edges from this new vertex to the three vertices on the boundary of the face, and assigning the labelx0_{to the new}

vertex in the triangular face bounded by the cycleuvwu, where{u

,

v

,

w

,

x} = {a

,

b

,

c

,

d}. Suppose we only use colors 1

,

2

,

3

,

4. Then it is clear from this construction thata,b,canddget different colors, and that the colors of a vertex and its primed counterpart are the same. Without loss of generality assume thataanda0_{get color 2. Then both}_b0_and_d_{must get color 4, a}

contradiction. It is routine to check that bbc2

(

G1

,

M

)

=5.

The following problems are still open.

Problem 18. Is bbc2

(

G

,

M

)

≤5 for any planar graphGwith a matching backboneM?

Problem 19. Is there a proof of bbc2

(

G

,

M

)

≤6 that does not require the Four Color Theorem? 7.2. Cyclic backbone colorings

In the last part of this section we introduce a special kind of 2-backbone coloring with a cyclic property as defined below. Our motivation for doing this is to get a better understanding of the structure of the original (acyclic) 2-matching backbone colorings of planar graphs. We prove a sharp result with respect to the upper bound on the number of colors needed to color planar graphs in the way explained below.

LetH=

(

V

,

EH)be a backbone of the graphG=

(

V

,

EG). A 2-backbone coloringf :V→ {1

, . . . , `

}of

(

G

,

H

)

is called an

`

-cyclic 2-backbone coloring of

(

G

,

H

)

, if no edge ofEHjoins two vertices with color 1 and color

`

inV. In a 2-backbone coloring

we say that two colorsxandyare adjacent if|x−y| ≤1. In an

`

-cyclic 2-backbone coloring we also say that color 1 and color

`

are adjacent.

The study of cyclic colorings in the context of frequency assignment is well motivated in [17]. For the proof ofTheorem 21we first construct the following useful gadget.

Lemma 20. LetHbe the graph with a matchingMconsisting of edgesab

,

cd

,

eu

,

f gandhias inFig. 5(a). LetGbe a graph with a matching backboneM0_{. If}_H _⊆ _G_and_M _⊆_M0_{, then vertex}_u_{and vertex}_v_{cannot be colored with two adjacent colors in a} 5-cyclic 2-backbone coloring of

(

G

,

M0

₎

_.

Proof. Suppose vertexuand vertexvcan be colored with two adjacent colors in a 5-cyclic 2-backbone coloring of

(

G

,

M0

)

_.

Since we use a 5-cyclic 2-backbone coloring, we can without loss of generality assume that vertexuis colored with color 1 and vertexvis colored with color 2. This leaves us with three possible colors for vertexd: color 3, color 4 or color 5.

• If vertexdis colored with color 3, then vertexemust get color 4. Continuing this way, vertexfgets color 5, vertexggets color 3 and vertexhgets color 4. Since there is no feasible color for vertexi, this implies a contradiction.

• If vertexdis colored with color 4, then vertexegets color 3, vertexf gets color 5, vertexggets color 3 and vertexhgets color 4. Again, we find a contradiction, since there is no feasible color for vertexi.

• If vertexdis colored with color 5, then vertexcmust get color 3 and the only feasible color for vertexbis color 4. We get a contradiction, since there is no feasible color for vertexa.

This completes the proof ofLemma 20.

Theorem 21. (a) LetGbe a planar graph with a matching backboneM. Then

(

G

,

M

)

has a 6-cyclic 2-backbone coloring. (b) There exist planar graphs that do not have a 5-cyclic 2-backbone coloring along a matching.