Ramsey Numbers for Line Graphs

Abstract

Given a graph, the classical Ramsey number R(k, l) is the least number of vertices that need to be in the graph for the existence of a clique of size k or an independent set of size l. Finding R(k, l) exactly has been a notoriously hard problem. Even R(k, 3) has not been determined for all values of k. Hence finding the Ramsey number for subclasses of graphs is an interesting question. It is known that even for claw-free graphs, finding Ramsey number is as hard as for general graphs for infinite number of cases. Line graphs are an important subclass of claw-free graphs. The question with respect to line graph L(G) is equivalent to the minimum number of edges the underlying graph G needs to have for the existence of a vertex with degree k or a matching of size l. Chvátal and Hanson determined this exactly for line graphs of simple graphs. Later Balachandran and Khare gave the same bounds with a different proof. In this paper we find Ramsey numbers for line graph of multi graphs thereby extending the results of Chvátal and Hanson. Here we determine the maximum number of edges that a multigraph can have, when its matching number, multiplicity, and maximum degree are bounded, and characterize such graphs.