Minimum Degrees of Minimal Ramsey Graphs

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 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; a graph H with s(H) = 2δ(H)−1 is called Ramsey simple. Szabó, Zumstein, and Zürcher were the first to ask which graphs are Ramsey simple, and conjectured that all bipartite graphs without isolated vertices are. Fox, Grinshpun, Liebenau, Person, and Szabó further conjectured that all triangle-free graphs without isolated vertices are Ramsey simple. We show that d-regular 3-connected triangle-free graphs, with one extra technical constraint, are Ramsey simple.