Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques

For graphs F and H, we say F is Ramsey for H if every 2-coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H-minimal if F is Ramsey for H and there is no proper subgraph F ′ of F so that F ′ is Ramsey for H. Burr, Erdős, and Lovász defined s(H) to be the minimum degree of F over all Ramsey H-minimal graphs F . Define Ht,d to be a graph on t + 1 vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that s(Ht,d) = d 2 for all values 1 < d ≤ t; it was previously known that s(Ht,1) = t− 1, so it is surprising that s(Ht,2) = 4 is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that s(H) ≥ 2δ(H)− 1 for all graphs H, where δ(H) is the minimum degree of H; Szabó, Zumstein, and Zürcher investigated which graphs have this property and conjectured that all bipartite graphs H without isolated vertices satisfy s(H) = 2δ(H) − 1. Fox, Grinshpun, Liebenau, Person, and Szabó further conjectured that all triangle-free graphs without isolated vertices satisfy this property. We show that d-regular 3-connected triangle-free graphs H, with one extra technical constraint, satisfy s(H) = 2δ(H)− 1.