The Minimum Degree of Ramsey-Minimal Graphs Jacob Fox and Kathy Lin DEPARTMENT OF MATHEMATICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02143 E-mail: licht@MIT.EDU kathylin@eden.rutgers.edu Received July 6, 2005; Revised May 29, 2006 Published online 16 November 2006 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20199 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(K k ) = (k - 1) 2 and s(K a,b ) = 2 min(a, b) - 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007 Keywords: Ramsey; critical; minimal; graph 1. INTRODUCTION In this article, we consider only finite simple graphs. The complete graph and the cycle on n vertices are denoted by K n and C n , respectively. We write H → (G; r) if every r-coloring of the edges of H contains a monochro- matic copy of G, and H → G if H → (G; 2). A graph H is G-minimal if H → G, but for every proper subgraph H ′ of H , H ′ → G. For example, K 6 is K 3 -minimal be- cause K 6 → K 3 , and no proper subgraph H ′ ⊂ K 6 has the property that H ′ → K 3 . The Ramsey number R(G) is the minimum positive integer n such that K n → G. In 1967, Erd˝ os and Hajnal [3] asked whether there exists a K 6 -free graph H such that H → K 3 . Graham [5] answered this question by showing that K 8 - C 5 Journal of Graph Theory © 2006 Wiley Periodicals, Inc. 167