The minimum degree of Ramsey-minimal graphs

Abstract

We write H → G if every 2-coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G-minimal if H → G, but for every proper subgraph H′ of H. H′ → G. We define s(G) to be the minimum s such that there exists a G-minimal graph with a vertex of degree s. We prove that s(Kk) = (k - 1)2 and s(Ka,b) = 2 min(a, b) - 1. We also pose several related open problems. © 2006 wiley Periodicals, Inc.