On characterizing the critical graphs for matching Ramsey numbers

Abstract

Given simple graphs H1,H2,…,Hc, the Ramsey number r(H1,H2,…,Hc) is the smallest positive integer n such that every edge-colored Kn with c colors contains a subgraph in color i isomorphic to Hi for some i∈{1,2,…,c}. The critical graphs for r(H1,H2,…,Hc) are edge-colored complete graphs on r(H1,H2,…,Hc)−1 vertices with c colors which contain no subgraphs in color i isomorphic to Hi for any i∈{1,2,…,c}. For n1≥n2≥⋯≥nc≥1, Cockayne and Lorimer (1975) showed that r(n1K2,n2K2,…,ncK2)=n1+1+∑i=1c(ni−1), in which niK2 is a matching of size ni. Using the Gallai–Edmonds Theorem, we characterized all the critical graphs for r(n1K2,n2K2,…,ncK2), implying a new proof for this Ramsey number.