The minimum degree of Ramsey‐minimal graphs

Abstract

AbstractWe 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. J Graph Theory 54: 167–177, 2007