The paper deals with graph operators—the Gallai graphs and the anti-Gallai graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs and the anti-Gallai graphs to be H-free for any finite graph H. The case of complement reducible graphs—cographs is discussed in detail. Some relations between the chromatic number, the radius and the diameter of a graph and it...

Given a family of (hyper)graphs F a (hyper)graph G is said to be F-saturated if G is F -free for any F ∈ F but for any edge e in the complement of G the (hyper)graph G + e contains some F ∈ F . We survey the problem of determining the minimum size of an F-saturated (hyper)graph and collect many open problems and conjectures.

Erdős and Hajnal conjectured that, for every graph H , there exists a constant cH such that every graph G on n vertices which does not contain any induced copy of H has a clique or a stable set of size nH . We prove that for every k, there exists ck > 0 such that every graph G on n vertices not inducing a cycle of length at least k nor its complement contains a clique or a stable set of size nk .

For two given graphs G1 and G2, the Ramsey number r(G1, G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Km denote a complete graph of order m and Kn −P3 a complete graph of order n without two incident edges. In this paper, we prove that r(K5 − P3,K5) = 25 without help of computer algorithms.

A graph G is called F -saturated if it does not contain any copy of F , but for any edge e in the complement of G the graph G + e contains some F . The minimum size of an n-vertex F -saturated graph is denoted by sat(n, F ). We give almost exact asymptotics for sat(n,Ck) as k is fixed and n → ∞ where Ck is a cycle with length k.

For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G),NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G),NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is d...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

A set of edges in a graph is sparse if no two of these edges belong to the same clique. It is shown here that if a graph has girth at least 5, 6 or 8 then the largest number of edges in a sparse set in its complement is at most 8, 7 or 6, respectively; this result is complete and best possible. It follows that if ¿ 0, then for su1ciently large n there exists a graph with n vertices and chromati...