The vertex independence number of a graph G is the maximal number of independent vertices in G. The clique number of G is the size of the largest complete subgraph of G. Let !1(v,n,r) denote the class of simple graphs on v vertices having vertex independence number n and clique number r. Let [(v,n,r) == min {dG): G E !1(v,n,r)}, where dG} denotes the number of edges in G, In this paper we study...

A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we ...

Let $M$ be an $R$-module and $0 neq fin M^*={rm Hom}(M,R)$. We associate an undirected graph $gf$ to $M$ in which non-zero elements $x$ and $y$ of $M$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. Weobserve that over a commutative ring $R$, $gf$ is connected anddiam$(gf)leq 3$. Moreover, if $Gamma (M)$ contains a cycle,then $mbox{gr}(gf)leq 4$. Furthermore if $|gf|geq 1$, then$gf$ is finit...

A multifamily set representation of a finite simple graph G is a multifamily F of sets (not necessarily distinct) for which each set represents a vertex in G and two sets in F intersects if and only if the two corresponding vertices are adjacent. For a graph G, an edge clique covering (edge clique partition, respectively) Q is a set of cliques for which every edge is contained in at least (exac...

Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw-free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw-free graph with chromatic number χ has a clique minor of size ⌈23χ⌉.