Some problems on chromatic polynomials

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 172 (1997) 39-44

Some problems on chromatic polynomials G.L. Chia

Department of Mathematics, Universityof Malaya, 59100 Kuala Lumpur, Malaysia Received 9 January 1995; revised 20 June 1995

Abstract In this paper, we describe some unsolved problems on chromatic polynomials along with a brief account of their progresses. Most of these problems are concerned with graphs that are uniquely determined by their chromatic polynomials. It is hoped that the solutions to these problems will uncover new methods and facts concerning chromatic polynomials.

Let P(G; 2) denote the chromatic polynomial of the graph G. Two graphs are said to be chromatically equivalent if they share the same chromatic polynomial. A graph is chromatically unique if it is chromatically equivalent only to itself. A graph is vertextransitive (respectively edge-transitive) if its group of automorphisms acts transitively on the vertex-set (respectively edge-set). A block is a connected nontrivial graph having no cut vertices. Let Kn denote the complete graph on n vertices. All undefined terms may be found in [15]. A number of people have proved the following result which answers a conjecture of Whitehead Jr. and Zhao [29] in the affirmative.

Theorem 1 (Chia [6], Read [24] and Xu [31]). Let G be a graph obtained by attaching a 1£2 to a block H. Then G is chromatically unique if and only if H is chromatically unique and vertex-transitive. Theorem 1 has led to the following problem.

Question 1. Which connected vertex-transitive graphs are chromatically unique? While disconnected vertex-transitive graphs (except for those having no edges) are easily seen not to be chromatically unique, not a great deal is known about the chromaticity for those vertex-transitive graphs that are connected. Some families of connected vertex-transitive graphs have been shown to be chromatically unique. Examples are the complete graphs Kn, cycles Cn [5], complete bipartite graphs Kn,n, complete t-partite graphs K2,2,...,2 [22], Kn........ [4], all connected vertex-transitive graphs with no 0012-365X/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0012-365X(96)00266-X

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G.L. ChialDiscrete Mathematics 172 (1997) 39-44

Fig. 1. more than 6 vertices [14] and all connected triangle-free vertex-transitive graphs with no more than 10 vertices [25]. In the meantime, some infinite families of connected vertex-transitive graphs that are not chromatically unique have also been found; see, for example, [8,12,20,24]. Let Vn denote the set of all connected vertex-transitive graphs on n vertices and CU~ the set of all connected vertex-transitive graphs on n vertices that are chromatically unique.

Question 2. Does lim,~o~

ICUnl/IV~[ exist?

I f so, what is the limit?

Theorem 1 has been generalized to q-graphs [3]. Extension of Theorem 1 to graphs of connectivity 2 has also been carried out [7,24]. A graph is quasi-separable if it contains a complete subgraph Kn such that G - K n is disconnected (see [2, p. 61]). Call such a Kn a separating complete subgraph. Note that in the event n ~<1, then G is separable. A quasi-block Q of G is a maximal subgraph of G that is not quasiseparable. Note that if n = 1, then Q is a block.

Question 3. Is it true that if G is chromatically unique, then G has at most two quasi-blocks? It is easy to see that if a graph G is chromatically unique, then G has at most two blocks (see [6]). This means that G has at most one cut vertex. For a graph with connectivity 2, Read [24] has shown that if it is chromatically unique, then it has at most one separating complete subgraph K2. Naturally, one would like to know if this is still true if K2 is replaced by Kn, n>~3? The graph in Fig. 1 (taken from [17]) is chromatically unique and has three separating complete subgraphs, a K3 and two Ka's. But here the/(3 and the two K4's are all in the complete subgraph/(5.

Question 4. Is it true that if G is chromatically unique, then either G has at most one separating complete subgraph, or else it has a complete subgraph Kn (n~>4) which contains all the separating complete subgraphs? Suppose G consists of two quasi-blocks Q1 and Q2 with Ql N Q2 ~ gn. Then G is said to have property ~ if for every i = 1, 2, there exists xi E Qi such that xi is

G.L. ChialDiscreteMathematics 172 (1997) 39-44 adjacent to all the vertices of Ql f3 Q2 = Kn and that N(x) denotes the neighborhood of x.

N(xi)

-

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gn is nonempty, where

Theorem 2 (Chia [7] and Read [24]). Let G be a graph consisting of two quasiblocks Q1 and Q2 with Q1 N Q2 = K2. Suppose G is chromatically unique. Then (i) G does not have property ~; (ii) Q1 and Q2 are chromatically unique; (iii) Q1 and Q2 are edge-transitive. Further, at least one of Q1 or Q2 is vertextransitive.

Question 5. ls the converse of Theorem 2 true? No counterexamples to Question 5 have been found. Results that support its truth can be found in [5,13,16]. Examples are the 0-graph and the graph obtained by overlapping Km and Cn at an edge. What seems to be the main hurdle to the solution of Question 5 is the lack of information that relates the multiplicity of the root ). --- 2 of the chromatic polynomial P(G; 2) and the structure of the graph G.

Question 6. What structural properties of G can be deduced from the multiplicity of the root 2 = 2 of the chromatic polynomial P(G; 2)? It is well known that the multiplicity of the root 2 = 0 characterizes the number of components in the graph. It has been proved (in [29,30]) that the multiplicity of the root 2 = 1 characterizes the number of blocks in the graph. Little is known about the case 2 = 2 except for some special cases given in [11,27,28]. A graph G is uniquely s-colorable if it has chromatic number s and any two s-colorings of G induce the same partition of its vertex-set into s independent subsets. These independent subsets are called the color classes of G. Note that if G is uniquely s-colorable, then P( G; s) = s!. Let n/> 1 and s ~>2 be positive integers and let S = {+1,-1-2 . . . . . + ( s -

1)}.

The graph G = G(ns, S) is defined to have the vertex-set

V(G) = {0, 1..... ns - 1} and edge-set E(G), where (i,j) E E(G)

if and only if i - j

ES

with the operation taken as modulo ns. Note that the graph G(ns, S) is uniquely s-colorable and vertex-transitive.

Question 7. Is it true that the graph G(ns, S) is chromatically unique?

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G.L. ChialDiscreteMathematics 172 (1997) 39-44

When n = 1, the graph G(ns, S) is the complete graph Ks and is chromatically unique. When n - - 2 , the complement of the graph G(ns, S) consists of s disjoint copies of K2. Loerinc and Whitehead Jr. [22] have proved that G(2s, S) is chromatically unique. For n/> 3, the answer is not known. Let F be a finite group and let H be a subset of F such that (i) the identity 1 ¢~H and (ii) h E H implies that h -1 E H. The Cayley graph X = Xr, I-i of F with respect to H is defined to have vertex-set V(X) = F and edge-set E(X) = {(g, gh): g E F, h E H} (see [2] for this definition). It is well known that a Cayley graph is vertextransitive. Question 8. What can be said about the chromaticity of Cayley graphs? The join of two graphs G and H, denoted by G+H, is defined to be the graph with vertex-set V(G + H) = V(G) U V(H) and edge-set E(G + H), where E(G + H) = E(G) U E(H) U {(x, y): x E V(G), y E V(H)}. Question 9. For what chromatically unique graphs G and H is G + H chromatically unique? The case when G is a complete graph Km has been considered in [8-10, 18,21]. An interesting problem is to determine the chromaticity of Cn + Cn. Note that Cn + Cn is vertex-transitive. Further, when n is even, it is uniquely 4-colorable. Question 10. Let G be a uniquely colorable graph & which each color class has exactly k vertices. When is G chromatically unique? Suppose G is a uniquely colorable graph in which each independent subset has exactly two vertices. In [8], it has been shown that for any m >~1, Km+ G is chromatically unique if and only if G is. Let G be a graph and let ~f(G) denote the set of all graphs having the same chromatic polynomial as that of G. Note that G is chromatically unique if Cg(G) -- {G}. Question 11. Suppose H is a chromatically unique graph and let G = H + ... + H be the join of m (~>2) copies of H. What is ~(G)? In [19], it has been shown that the complement of any 2-regular graph containing no C3 or Ca is chromatically unique. This result implies that Cn + Cn "~-"'" + Cn is chromatically unique if n ~>5, where G denotes the complement of the graph G. Question 12. What is the chromatic polynomial for the graph G × H, the cartesian product of the graphs G and H?

G.L. ChialDiscrete Mathematics 172 (1997) 39~4

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The chromatic polynomial for Cn × K2 has been found [2]. Let L(r, s, n) denote the total number of Latin rectangles of order r x s based on an n-set. In [1], it has been shown that P(L(Kr,~);n) = L(r,s,n), where L(Kr, s) is the line graph of the complete bipartite graph Kr,~. The authors have also obtained the chromatic polynomials for L(K3,s) and L(Ka, s) and hence those of K3 x Ks and K4 x Ks because L(Kr, s) = Kr x Ks.

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[29] E.G. Whitehead Jr. and L.-C. Zhao, Cutpoints and the chromatic polynomial, J. Graph Theory 8 (1984) 371-377. [30] D.R. Woodall, Zeros of chromatic polynomials, in: P.J. Cameron, ed., Combinatorial Survey, Proc. Sixth British Combinat. Conf. (Academic Press, New York, 1977) 199-223. [31] S.-J. Xu, Some notes on chromatic uniqueness of graphs, J. Shanghai Teach. Univ., Nat. Sci. Ed. 2 (1987) 10-12 (Chinese, English summary).