  A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs
$(H_1,\ldots,H_q)$, denoted by $G\to_q(H_1,\ldots,H_q)$, if every
$q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$
for some $i\in[q]$. 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, Erd\H{o}s and Lov\'asz, 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 $H_i\cong H$ for all $i\in [q]$. 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.
