A long-standing conjecture in combinatorics, made by Erdős and Simonovits, is that the maximum number of edges in an n-vertex graph without a hexagon is asymptotically 1 2n 4/3 as n→∞. This conjecture corresponds to the asymptotic optimality of constructions of generalized quadrangles as a source of dense hexagon-free graphs. In this paper, we construct a counterexample to this conjecture. For ...

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, an...

A graph G is triangularly connected if for every pair of edges e1, e2 ∈ E(G), G has a sequence of 3-cycles C1, C2, · · · , Cl such that e1 ∈ C1, e2 ∈ Cl and such that E(Ci) ∩ E(Ci+1) 6= ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former results in [2], [3], [4] and [5] that every connecte...

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no K3,3 minor, and consequently for planar graphs. Carsten Thomassen made the following conjecture [1, 7]: Conjecture 1 (Thomassen) Every longest circuit of a 3-connected graph has a chord. That conjecture has been proved for planar graphs with minimum degree at...

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let K4 and K1,3 (claw) denote the complete (bipartite) graph on 4 and 1+ 3 vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a K4-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free gr...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Graph grammars have been studied by several authors. The motivations of these works have arisen from completely different areas of computer science, such as picture processing (Pfaltz and Rosenfeld, 1969; Montanari, 1970; Rosenfeld and Milgram, 1972; Pavlidis, 1972; Abe et al., 1973; Furtado and Mylopoulos, 1975; Fu and Brayer, 1975) or formal semantics of programming languages (Pratt, 1971). I...

We prove that every 2{connected K 1;3-free and Z 3 ?free graph is hamiltonian except for two graphs. Furthermore, we give a complete characterization of all 2?connected, K 1;3-free graphs, which are not pancyclic, and which are Z 3-free, B-free, W-free, or HP 7 ?free.