On upper domination Ramsey numbers for graphs

Abstract

The classical Ramsey number r( m, n) can be defined as the smallest integer p such that in every two-coloring ( R, B) of the edges of K p , β( B)⩾ m or β( R)⩾ n, where β( G) denotes the independence number of a graph G. We define the upper domination Ramsey number u( m, n) as the smallest integer p such that in every two-coloring ( R, B) of the edges of K p , Γ( B)⩾ m or Γ( R)⩾ n, where Γ( G) is the maximum cardinality of a minimal dominating set of a graph G. The mixed domination Ramsey number v( m, n) is defined to be the smallest integer p such that in every two-coloring ( R, B) of the edges of K p , Γ( B)⩾ m or β( R)⩾ n. Since β( G)⩽ Γ( G) for every graph G, u( m, n)⩽ v( m, n)⩽ r( m, n). We develop techniques to obtain upper bounds for upper domination Ramsey numbers of the form u(3, n) and mixed domination Ramsey numbers of the form v(3, n). We show that u(3,3)= v(3,3)=6, u(3,4)=8, v(3,4)=9, u(3,5)= v(3,5)=12 and u(3,6)= v(3,6)=15.