A note on group colorings

A Note on Group Colorings

Daniel Kral’,1 Ondrej Pangrac,2 and Heinz-Jurgen Voss3

1INSTITUTE FOR THEORETICAL COMPUTER SCIENCE (ITI)

FACULTY OF MATHEMATICS AND PHYSICS

CHARLES UNIVERSITY MALOSTRANSKE NAMESTI 25

118 00 PRAGUE, CZECH REPUBLIC

E-mail: [email protected]

2DEPARTMENT OF APPLIED MATHEMATICS AND

INSTITUTE FOR THEORETICAL COMPUTER SCIENCE (ITI)

FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY

MALOSTRANSKE NAMESTI 25

118 00 PRAGUE, CZECH REPUBLIC

E-mail: [email protected]

3INSTITUTE OF ALGEBRA

TECHNICAL UNIVERSITY DRESDEN, GERMANY

Received March 1, 2004; Revised February 2, 2005

Published online 6 June 2005 in Wiley InterScience(www.interscience.wiley.com).

DOI 10.1002/jgt.20098

Abstract: We study a concept of group coloring introduced by Jaegeret al. We show that the group chromatic number of a graph with minimumdegree � is greater than �=(2 ln �) and we answer several open questions onthe group chromatic number of planar graphs: a construction of a bipartiteplanar graph with group chromatic number four and a 3-colorable planargraph with group chromatic number five are presented. We also observe

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Contract grant sponsor: Ministry of Education of the Czech Republic as projectLN00A056 (to Institute for Theoretical Computer Science).Daniel Kral’ has been a postdoctoral fellow at Institute for Mathematics of theTechnical University Berlin since October 2004.

� 2005 Wiley Periodicals, Inc.

123

that several upper bounds on the choice number for various subclasses ofplanar graphs also translate to the concept of group colorings.� 2005 Wiley Periodicals, Inc. J Graph Theory 50: 123–129, 2005

Keywords: graph coloring; group coloring; nowhere-zero flows

1. INTRODUCTION

Group colorings of graphs have been introduced by Jaeger, Linial, Payan, and

Tarsi [3]. For plane graphs, this concept is dual to group connectivity. For an

Abelian group A, a graph G is said to be A-colorable if for every orientation ~GG of

G and for every ’ : Eð~GGÞ ! A, there is a vertex coloring c : VðGÞ ! A such that

cðwÞ � cðvÞ 6¼ ’ðvwÞ for each vw 2 Eð~GGÞ. Such a coloring c is called proper.

Clearly, the choice of an orientation ~GG is unimportant. A graph G is A-connected,

if for every orientation ~GG and for every ’ : Eð~GGÞ ! A, there is an A-flow f of ~GGsuch that f ðvwÞ 6¼ ’ðvwÞ. A plane graph is A-colorable if and only if its dual is

A-connected [3]. It is unknown whether the property of being A-colorable (A-

connected) depends on the structure or only on the order of A. The least number

�gðGÞ such that G is A-colorable for each Abelian group A of order at least �gðGÞis the group chromatic number of G.

Lai and Zhang [5] proved that the group chromatic number of a K5-minor free

graph is at most five using techniques of [7,9] developed for choosability. Thus,

�gðGÞ � 5 for each planar graph G. We shall construct a 3-colorable planar G

such that �gðGÞ ¼ 5. This answers a question from [3] requesting the deter-

mination of the group chromatic number for planar graphs. Several other

techniques developed for choosability also translate to the group coloring: planar

graphs without 3-cycles and 4-cycles have the group chromatic number at most

three [8] and planar graphs without 4-cycles at most four [6]. We do not provide

translated proofs and the reader is asked to verify the details. Similarly as in the

case of choosability [10,11], the bounds are best possible for group coloring, too.

The different concepts of group coloring and list coloring seem to be

intimately related: a graph has a large group chromatic number if it contains a

dense subgraph (with average degree �ð�g ln�gÞ, the corollary after Theorem 1)

similarly as in the case of choosability [1] (where an exponential average degree

is needed). From this point of view, we find interesting that each bipartite planar

graph is 3-choosable [2], but there is a bipartite planar graph with the group

chromatic number four (Theorem 2). On the other hand, �gðGÞ � 4 for each

bipartite planar graph G because each d-degenerate graph has the group

chromatic number at most d þ 1 [3,4].

The dual forms of our results are also of some interest: there is a planar 4-

regular 4-edge-connected graph that is not Z3-connected (such a non-planar graph

was constructed in [3]) and a 3-edge connected (bipartite) planar graph that is

neither Z4 nor Z2 � Z2-connected. Our constructions witness that the bounds in

Conjecture 4 and Conjecture 5 stated in [3], if true, would be best possible for

124 JOURNAL OF GRAPH THEORY

planar graphs, too. Conjecture 5 from [3] is actually a strengthening of Tutte’s 5-

flow conjecture.

2. A LOWER BOUND ON THE GROUP CHROMATIC NUMBER

We show that a graph has a large group chromatic number if it contains a dense

subgraph:

Theorem 1. If G is a graph with minimum degree � � 2, then �gðGÞ >�=ð2 ln �Þ.

Proof. Fix an Abelian group A of order k � �=ð2 ln �Þ and let a1; . . . ; ak be

the elements of A. We show that the graph G is not A-colorable. Let n and m be

the order and the size of G, respectively, and let e1; . . . ; em be the edges of G.

Note that m � n�=2.

Orient edges e1; . . . ; em of G arbitrarily. Let �0 be the set of all kn possible

vertex-colorings of G by the elements of A. We construct inductively a function

’ : EðGÞ ! A such that the graph G does not have a proper coloring with respect

to ’ and the fixed orientation of G. Consider i, 1 � i � m, and let ei ¼ viwi. Let

�j, 1 � j � k, be the number of colorings c of �i�1 such that aj ¼ cðwiÞ � cðviÞ.We set ’ðviwiÞ to be the most common difference aj0 and �i to be the set of the

colorings c of �i�1 with cðwiÞ � cðviÞ 6¼ ’ðviwiÞ. Note that �j0 � j�i�1j=k.

Therefore, we can conclude that j�ij � ð1 � 1=kÞj�i�1j and j�mj � ð1 � 1=kÞmkn.The set �m contains all the colorings of vertices of the graph G, which are

proper with respect to the fixed orientation and the function ’. However, the set

�m is empty:

j�mj � ð1 � 1=kÞmkn � e�m=ken ln k

� e� n�

2�=ð2 ln �Þþn ln �2 ln � ¼ en ln �

2 ln ��ln �ð Þ ¼ e�n ln ð2 ln �Þ < 1: &

Corollary. If a graph G contains a subgraph with minimum degree � � 2, then

�gðGÞ > �=ð2 ln �Þ.

3. A BIPARTITE PLANAR GRAPH G WITH ��g(G)¼ 4

In this section, we construct a bipartite planar graph with group chromatic number

four:

Theorem 2. There is a bipartite planar graph whose group chromatic number

is 4.

Proof. We construct a graph G, which is not Z3-colorable. At the same time,

we provide its orientation and a ‘‘bad’’ edge-labeling. The graph G is obtained by

pasting the gadget drawn in the right part of Figure 1 to each of the zones A, B,

A NOTE ON GROUP COLORINGS 125

and C of the skeleton in the left part. The vertex a of the gadget is identified with

the vertex �, the vertex c with �, and the vertices b and d with the degree-two

vertices of the skeleton. Thus, the dashed edges become identical. The edge

labels (a function ’) are those depicted in the figure (the missing labels depend

on the zone and are described later). Since each face of G has size four, G is

bipartite.

If there is a Z3-coloring c of G, we can assume that cð�Þ ¼ 0. If cð�Þ ¼ 1, then

in the gadget pasted into the zone A, the coloring cðbÞ ¼ 2 and cðdÞ ¼ 1 is forced.

If cð�Þ ¼ 2, the same holds for the zone B and if cð�Þ ¼ 0, for the zone C. We

describe and analyze only the gadget of the zone A (the case cð�Þ ¼ 1); the other

two cases are analogous.

Note that cðaÞ ¼ 0, cðbÞ ¼ 2, cðcÞ ¼ 1, and cðdÞ ¼ 1 for the zone A. The label

(missing in the figure) of the edge incident with the vertex a of the gadget is 2, of

the edge incident with b is 0, of the edge incident with c is 1, and the labels of the

edges incident with d are both 1. This forces that the vertices of the inner hexagon

adjacent to the vertices a, b, c, and d can be colored only by 0 and 1. Because of

the edges with label 0, the three vertices depicted by empty circles must have the

same color. But then the vertex x adjacent to all the three vertices can be colored

neither by 0, 1, nor 2. Thus, the gadget pasted to the zone A forces that cð�Þ 6¼ 1.

Similarly, the other two gadgets forbid the remaining two possible values. &

4. A PLANAR GRAPH G WITH ��g(G)¼ 5

In this section, we construct a planar graph whose group chromatic number is

five:

Theorem 3. There is a 3-colorable planar graph whose group chromatic

number is 5.

Proof. We present a construction of a graph G and show that it is not Z4-

colorable. The arguments work smoothly for Z2 � Z2-coloring, too. First, two

copies of the right gadget depicted in Figure 2 are pasted to the zones I and II of

FIGURE 1. A construction of a bipartite planar graph G with �g(G )¼ 4.


the middle gadget. The vertices b and c of the right gadget coincide with their

counterparts of the middle one and the vertex a with the vertex a or d depending

on the zone to which the gadget is pasted in. The dashed edges become identical.

The graph G is obtained by pasting the middle gadget to each of the zones A,

B, C, and D of the skeleton in its left part. The vertex a of the middle gadget is

identified with the vertex �, the vertex d with �, and the vertices b and c with the

middle degree-three vertices of the skeleton. Again, the dashed edges become

identical. The obtained graph G can be found in Figure 3. Note that G is 3-

colorable because it is a plane triangulation whose all vertex degrees are even.

We show that there is no Z4-coloring regardless the orientation and the labels

of the dotted edges (therefore, we do not fix them). If there is a Z4-coloring c of

G, it can be assumed that cð�Þ ¼ 0. Each of the gadgets of the zones A, B, C, and

D forbids a single group element to be assigned to �. We describe the labels of the

middle gadget forbidding cð�Þ ¼ 0 (depicted in the middle of Fig. 2). The other

cases are symmetric. Consider the orientation and the edge labeling as in the

figure and assume cð�Þ ¼ cðdÞ ¼ 0. The vertices b and c can be colored only by 0

and 1. Note that cðbÞ 6¼ cðcÞ because of the edge between b and c. The gadgets in

the zones I and II are oriented and edge-labeled in such a way that one of them

forbids the case cðbÞ ¼ 0 and cðcÞ ¼ 1 and the other the case cðbÞ ¼ 1 and

cðcÞ ¼ 0. We analyze the gadget for the zone I that forbids the former.

Consider the orientation and the edge-labeling as in Figure 2. The edges

incident with the vertices a, b, and c force the three vertices of the middle triangle

FIGURE 2. A construction of a 3-colorable planar graph G with �g(G )¼ 5.

FIGURE 3. The graph G constructed in the proof of Theorem 3.


(depicted by empty circles) to be colored only by 0 and 1. But two group

elements are not enough to color these three vertices. Since the other gadgets

prevent assigning the remaining three elements to �, G has no Z4-coloring. &

5. CONCLUSION

As noted in Introduction, the concepts of group coloring and list coloring seem to

be related: dense subgraphs force both the group chromatic number and the

choice number to be large, but the group chromatic number seems to grow much

faster, e.g., the group chromatic number of even cycles is three [4] but their

choice number is only two [1]. Let us remind that the choice number �lðGÞ of a

graph is the least integer k such that the graph G can be colored from any lists of

sizes k. It seems plausible to pose the following conjecture:

Conjecture 1. The following holds for each graph G:

�lðGÞ � �gðGÞ:

In other words, the choice number �lðGÞ does not exceed the group chromatic

number �gðGÞ for any graph G.

We have recently learned that Conjecture 1 has been made independently of us

by Margit Voigt, and it was around even before, in particular among the

researchers at West Virginia University, as pointed to us by one of the referees.

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[4] H.-J. Lai and X. Zhang, Group colorability of graphs, Ars Combin 62 (2002),

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[5] H.-J. Lai and X. Zhang, Group chromatic number of graphs without K5-

minors, Graphs and Combinatorics 18 (2002), 147–154.

[6] P. C. B. Lam, The 4-choosability of plane graphs without 4-cycles, J Combin

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[7] R. Skrekovski, Choosability of K5-minor free graphs, Discrete Math 190

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[8] C. Thomassen, 3-list-coloring planar graphs of girth 5, J Combin Theory Ser

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A note on group colorings