On mixed ramsey numbers

Abstract

For positive integers m and n the classical ramsey number r( m, n) is the least positive integer p such that if G is any graph of order p then either G contains a subgraph isomorphic to K m or the complement G of G contains a subgraph isomorphic to K n . Some authors have considered the concept of mixed ramsey numbers. Given a graph theoretic parameter f, an integer m and a graph H, the mixed ramsey number v( f; m; H) is defined as the least positive integer p such that if G is any graph of order p, then either f(G) ⩾ m or G contains a subgraph isomorphic to H. In this paper we consider the problem of determining the mixed ramsey numbers for vertex linear arboricity and some other generalizations of chromatic number. We discuss the above problem for various structures H such as the complete graph, the claw, the path and the tree. Further, we study the generalized mixed ramsey number v( f; m 1, m 2,…, m 1; H l + 1 , H l + 2 ,…, H k ), where the edge set of the complete graph is partitioned into k sets.