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, Erdos, and Lovasz 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)=d2
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; Szabo, Zumstein, and Zurcher 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 Szabo 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; the extra constraint is that H has a vertex v so that if one
removes v and its neighborhood from H, the remainder is connected.Comment: 10 pages; 3 figure