A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,ldots,H_q)$, denoted by $Gto_q(H_1,ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $iin[q]$. Let $s_q(H_1,ldots,H_q)$ denote the smallest minimum degree of $G$ over all graphs $G$ that are minimal $q$-Ramsey for $(H_1,ldots,H_q)$ (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, ErdH{o}s and Lovasz, who determined its value precisely for a pair of cliques. Over the past two decades the parameter $s_q$ has been studied by several groups of authors, the main focus being on the symmetric case, where $H_icong H$ for all $iin [q]$. 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 $s_2(H_1,H_2)$ when $(H_1,H_2)$ is a pair of one clique and one tree, a pair of one clique and one cycle, and when it is a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on $s_q(C_ell,ldots,C_ell,K_t,ldots,K_t)$ 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.
تحميل البحث