Classes of chromatically unique graphs

Borowiecki, M. and E. Drgas-Burchardt, Classes of chromatically unique graphs, Discrete Mathematics Ill (1993) 71-75. We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique. 1. Preliminary definitions and results In this paper we consider finite, undirected, simple and loopless graphs. Two graphs G and H are said to be chromatically equivalent if they have the same chromatic polynomial, i.e. PH(j_)=PG(i). A graph G is said to be chromatically unique if P,(i)=P,(;I) implies that H is isomorphic to G. At present, any method for recognizing whether a given graph is chromatically unique is not known. For large graphs, it is extremely difficult to prove chromatic uniqueness. Some classes of chromatically unique graphs are well known. See, for instance, [ 1,2,5-71 and their references, where some families of such graphs are presented. Here, by solving an extremal problem for bipartite graphs, we prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique. By a bipartite graph we mean such a graph whose vertex set can be partitioned into two nonempty sets U and V, called colour classes, such that every edge of the graph Correspondence to: M. Borowiecki, Institute of Math., Higher College of Engineering, Podgrona 50, 65-246 Zielona Gora, Poland. 0012-365X/93/$06.00 ‘c 1993-Elsevier Science Publishers B.V. All rights reserved 12 M. Borowiecki, E. Drgas-Burchardt joins an element of U with an element of I/. If colour classes of a bipartite graph have the same cardinality then we call it equibipartite. The reader is referred to [4] for further information. Using the chromatic polynomial, the bipartite graphs can be characterized as follows. Proposition 1.1. G is bipartite graph if and only if PC(A) is not divisible by A2. From this we immediately have the following proposition. Proposition 1.2. If H is chromatically equivalent to G and G is bipartite, then H is bipartite. Thus, the only candidates to be chromatically equivalent to a bipartite graph are bipartite graphs. According to [3], for a given graph G, the first four coefficients of its chromatic polynomial can be written as follows: + [ 0 ; +(q-2)N,(G)+N,(G)-2N,(G) 1 A”-‘+ ..., where p, q, N,(G), Np(G) and N,(G) denote the number of vertices, edges, triangles (cycles of order three), pure quadrilaterals (cycles of order four without chords) and complete graphs with four vertices of G, respectively. For a bipartite graph G, we have N,(G)=N,(G)=O. Thus, we have the following result. Proposition 1.3. If H is chromatically equivalent to a bipartite graph G, then H must be a bipartite graph having the same number of vertices, edges and pure quadrilaterals as G. Let G = ( U, V; E) be a bipartite graph with 1 U I= m and ) V/ = n. Such a graph will also be denoted by G,,, to emphasize the cardinalities of colour classes of G. Without loss of generality, we assume that m < n. For a given graph G,,, with U= {ui, . . . . u,}, V= {vi, . . . , vn), let us denote by X(G,,,) the m x n matrix [xij] defined by Xii= 1 if there exists an edge joining Ui and vj, and xij = 0 otherwise. Thcdegrees of ui and vj are denoted by di and ej, respectively. The complementary graph G,+, of G,,. with respect to K,,, is defined by its matrix X(6,,,,)= [gij] in the following way: Ziij= 1 -xij. Degrees of vertices of G,,, will be denoted by LZ$ and Zj, respectively. Classes of chromatically unique graphs 73 Proposition 1.4 Salzberg et al. [6]. NQ(G,,,)=+n+nl)(nl)-&[(2m1)(2nl)q -(2ml)D”-(2n1$+4&s”-24’1, where 4” denotes the number of edges of c?,,,, c=~y=i d”?, E”=cJ=i Zf, R”=c~;“&Ej’jx”ij, s=&j,j, j$jj&j,xi,jxi,j,. 2. The chromatic uniqueness of K,,,--rKz The graph G=K,_-rK2, 0 d r bm, has order 2m and size m2 -r. According to Proposition 1.3, a prospective candidate to be chromatically equivalent to G should be a bipartite graph G,,, satisfying s+t=2m and st am’-r. By solving these two expressions for s or t, we obtain m$ds<t<m+J. Parametrizing s and t, we have s=mk, t=m+ k for 0~ k<J. This implies the following result. Lemma 2.1. If H is chromatically equivalent to G= K,,,-rK2, then H= Hm_k,m+k for some k, Odk<$. H can be obtained from the complete bipartite graph Km_k,m+k by deleting r k2 edges. Lemma 2.2. Let G=K,,,-rK,. Then N,(G)=h[m’(m-l)*-2r(2m2-4m+3-r)]. Proof. For the graph c”, we have 4” = fi= E”= R”= f= r. By Proposition 1.4 and simple calculations we have the require formula. 0 Let us denote by S; the graph Km_k,m+k-(r-k2)Kfr where O<k<$. It is easy to see that S; is a candidate to be chromatically equivalent to G = K,,, -rK2. Lemma 2.3. N,(S~)=~[(m2-k2)((m-1)2-kk2)-2(r-k2)(2(m2-k2)-4m+3-(r-k2))]. Proof. By the fact that q=D”=E”= l?=S=r-k* for Fk and Proposition 1.4, the formula follows. 0 Lemma 2.4. Let G = K,,, rK2. Then for k 2 1 and m 2 3,