Critical graphs for clique-coloring

Meziane Aider, a Sylvain Gravier b

aInstitut de Mathematiques, USTHB BP 32 El Alia, 16 111 Bab Ezzouar, Alger,

Algeria, E-mail : meziane.aider@imag.fr

, bCNRS, Laboratoire Leibniz, 46 avenue Felix Viallet, 38031 Grenoble Cedex,

France, E-mail :sylvain.gravier@imag.fr

.

Abstract

We consider the NP-complete problem of coloring the vertices of a graph so that

no maximal clique of size at least two is monocolored. A basic property of graph

colorations is that a coloration of a graph is also a good coloration of all subgraphs.

This property allows to de�ne various notions of `critical graphs'. Nothing like these

is true for clique-colorations: �rst of all, a clique-coloration of G is usually not a

clique-coloration of the subgraphs. In this note, we propose some alternative de�ni-

tions of critical-graph for clique-coloration and we give some properties and families

of critical-graphs.

Key words: Graph coloring, clique coloring, clique-chromatic number, critical

graphs.

1 Introduction

A hypergraph is a pair H = (V; E) where V is the set of vertices of H and E is

a family of non-empty subsets of V called edges. A k-coloring of H = (V; E)

is a mapping c : V ! f1; 2; : : : ; kg such that for all e 2 E with jej � 2 there

exist u; v 2 e with c(u) 6= c(v), i.e., there is no monocolored edge of size atleast two. The chromatic number of H is the smallest k such that H has a

k-coloring.

Here we consider hypergraphs arising from graphs: for a given graph G =

(V;E), the clique-hypergraph of G is the hypergraph H(G) = (V; E) whosevertices are the vertices of G and whose edges are the maximal cliques of

G. (A subset K � V is a clique of G if any two vertices of K are adjacent;

K is a maximal clique if it is not properly contained in another clique.) The

Preprint submitted to Elsevier Preprint 15 June 2000

coloring problem of such hypergraphs (henceforth \clique-hypergraph coloring

problem") was recently studied in [1], [4] and [5]. For convenience, we denote

�(G) as the chromatic number of H(G), and �G as the complement of G.

2 Preliminary

As noticed in [1], the basic hereditary properties of graph coloring can not

be extended to clique-coloring. Indeed, for any k > 2, consider the graph G

obtained from a k-clique-chromatic graph G0 by adding a vertex v which is

adjacent to all vertices of G0. Clearly, G is 2-clique-colorable (color 1 the vertex

v and 2 all the others). Nevertheless, G�v = G0 is not (k�1)-clique-colorable.

As shown in the following proposition, if we remove an edge e to a graph G

then �(G�e) is bounded by �(G). Hence, this allows to de�ne critical graphs.

Proposition 1 For any edge e of a graph G, we have

�(G)� 1 � �(G� e) � �(G) + 1.

By Proposition 1, we also have :

Proposition 2 For any edge e of the complement of a graph G, we have

�(G)� 1 � �(G + e) � �(G) + 1.

From Propositions 1 and 2, we de�ne four notions of critical graph :

� A (+1; remove)-critical graph is a graph G which satis�es �(G�e) = �(G)+

1 for every edge e of G,

� A (�1; remove)-critical graph is a graphG which satis�es �(G�e) = �(G)�

1 for every edge e of G,

� A (+1; add)-critical graph is a not complete graph G which satis�es �(G+

e) = �(G) + 1 for every edge e of �G,

� A (�1; add)-critical graph is a not complete graph G which satis�es �(G+

e) = �(G)� 1 for every edge e of �G.

In the next section, we exhibit some families of critical graphs for each de�ni-

tion, when such graphs exist.

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3 Critical graphs

In this section, we prove that in two cases, the notion of critical graphs is empty

in the sense that either there is no such graphs or there is no non trivial such

graphs. These cases are those for which the clique-chromatic number has to

increase by one unit. In the two other cases, we construct some classes of

critical graphs.

Proposition 3 There is no (+1; remove)-critical graph.

Proposition 4 A (+1; add)-critical is a stable of size at least two.

Observe that any triangle-free k-chromatic graph G such that G � e is (k �

1)-colorable is a (�1; remove)-critical graph since in this case H(G) = G.

So it is probably very hard to characterize (�1; remove)-critical graphs. Our

results now concern (�1; add)-critical graphs. Consider the well-known class

of triangle-free graphs de�ned by Mycielski [7] :

{ G2 consists on two adjacent vertices,

{ for any k > 2, the graph Gk = (Vk; Ek) is de�ned by :

� Vk = Vk�1[Sk[fxkg, where Vk�1 = fv1; : : : ; vnk�1g and Sk = fs1; : : : ; snk�1g,

� the subgraph induced by Vk�1 is isomorphic to Gk�1, and the subgraph

induced by Sk is a stable set,

� there exists an edge sivj if and only if there exists an edge vivj,

� xk is adjacent to all vertices in Sk and to no other vertex.

It is easy to show by induction that Gk is triangle-free, and �(Gk) = k, for all

k � 2.

Theorem 5 For any k > 2, the graph Gk is both a (�1; remove)-critical and

(�1; add)-critical graph.

References

[1] G. Bacs�o, S. Gravier, A. Gy�arf�as, M. Preissmann, A. Seb}o, Coloring the maximalcliques of graphs, Manuscript (1999).

[2] P. Erd}os, A. Hajnal, On the chromatic number of graphs and set systems, ActaMath. Acad. Sci. Hungar. 17 (1966), 61-99.

[3] A. Gy�arf�as, Problems from the world surrounding perfect graphs, ZastosowaniaMatematyki - Applicationes Mathematicae vol. XIX 3-4 (1987), 413-441.

[4] C. Ho�ang, S. Gravier, F. Ma�ray, Coloring the hypergraph of maximal cliquesof a graph with no long path, Submitted (1999).

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[5] C. Ho�ang, C. McDiarmid, On the divisibility of graphs, Lakehead University,Ontario, Technical report 98-10-1, 1998.

[6] F. Ma�ray, M. Preissmann, On the NP-completeness of the k-colorabilityproblem for triangle-free graphs, Discrete Mathematics 162 (1996), 313-317.

[7] J. Mycielski, Sur le coloriage des graphes, Colloq. Math. 3 (1955), 161{162.

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