For terms not defined here, the reader is referred to [6]. For a graph G=(V, E), we denote by G the complement of G. We use |(G) to denote the size of a largest clique in G and :(G) to denote the size of a largest stable set in G, or simply | and : when no confusion is possible. A graph G is said to be perfect if, for each induced subgraph H of G, H can be coloured with |(H) colours such that e...

We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let G be a graph and let G be its complement, then given the chromatic polynomial of G, we give a r...

For a graph G, let P(G; ) be its chromatic polynomial and let [G] be the set of graphs having P(G; ) as their chromatic polynomial. We call [G] the chromatic equivalence class of G. If [G]={G}, then G is said to be chromatically unique. In this paper, we 4rst determine [G] for each graph G whose complement 5 G is of the form aK1∪bK3∪⋃16i6s Pli , where a; b are any nonnegative integers and li is...

How few edge-disjoint triangles can there be in a graph G on n vertices and in its complement G? This question was posed by P. Erdó́s, who noticed that if G is a disjoint union of two complete graphs of order n=2 then this number is n=12 þ o(n). Erdó́s conjectured that any other graph with n vertices together with its complement should also contain at least that many edge-disjoint triangles. In t...

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and on the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, relations of a similar type have been proposed for many other graph invariants, in several hundred papers. We present a survey on this research endeavor.

If G is a maximal exceptional graph then either (a) G is the cone over a graph switching-equivalent to the line graph L(K8) or (b) G has K8 as a star complement for the eigenvalue −2 (or both). In case (b) it is shown how G can be constructed from K8 using intersecting families of 3-sets.

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given gr...