The interval number of dense graphs

The Ramsey number of dense graphs

The Ramsey number r(H) of a graphH is the smallest number n such that, in any two-colouring of the edges of Kn, there is a monochromatic copy of H . We study the Ramsey number of graphs H with t vertices and density ρ, proving that r(H) ≤ 2 √ ρ . We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree ρt and the Ramsey number of random g...

On the interval number of special graphs

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i (G ). Griggs and West showed that i(G) d12(d þ 1)e. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independe...

The Domatic Number Problem in Interval Graphs

Rainbow connection number of dense graphs

An edge-colored graph G is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ ( n−2 2 ) + 2, and rc(G) ≤ 4 if |E(G)| ≥ ( n−3 2 ) + 3. These bounds...

On the fixed number of graphs

‎A set of vertices $S$ of a graph $G$ is called a fixing set of $G$‎, ‎if only the trivial automorphism of $G$ fixes every vertex in $S$‎. ‎The fixing number of a graph is the smallest cardinality of a fixing‎ ‎set‎. ‎The fixed number of a graph $G$ is the minimum $k$‎, ‎such that ‎every $k$-set of vertices of $G$ is a fixing set of $G$‎. ‎A graph $G$‎ ‎is called a $k$-fixed graph‎, ‎if its fix...