On the minimum degree of minimal Ramsey graphs for cliques versus cycles

A graph G is said to be q-Ramsey for a q-tuple of graphs (H₁,..., H_q), denoted by G →q (H₁,..., H_q), if every q-edge-coloring of G contains a monochromatic copy of Hi in color i for some i ε [q]. Let sq(H1,..., Hq) denote the smallest minimum degree of G over all graphs G that are minimal q-Ramsey for (H₁,..., H_q) (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, Erdos, and Lovasz, who determined its value precisely for a pair of cliques. Over the past two decades the parameter sq has been studied by several groups of authors, their main focus being on the symmetric case, where Hi ≅ H for all i ε [q]. The asymmetric case, in contrast, has received much less attention. In this paper, we make progress in this direction, studying asymmetric tuples consisting of cliques, cycles, and trees. We determine s₂(H₁, H₂) when (H₁, H₂) is a pair of one clique and one tree, a pair of one clique and one cycle, and a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on s_q(Cℓ,..., Cℓ, K_t,..., K_t) in terms of the size of the cliques t, the number of cycles, and the number of cliques. Our bounds are tight up to logarithmic factors when two of the three parameters are fixed.