Chvátal-type results for degree sequence Ramsey numbers

Abstract: A sequence of nonnegative integers $\pi =(d_1,d_2,...,d_n)$ is graphic if there is a (simple) graph $G$ of order $n$ having degree sequence $\pi$. In this case, $G$ is said to realize or be a realization of $\pi$. Given a graph $H$, a graphic sequence $\pi$ is potentially $H$-graphic if there is some realization of $\pi$ that contains $H$ as a subgraph. In this paper, we consider a degree sequence analogue to classical graph Ramsey numbers. For graphs $H_1$ and $H_2$, the potential-Ramsey number $r_{pot}(H_1,H_2)$ is the minimum integer $N$ such that for any $N$-term graphic sequence $\pi$, either $\pi$ is potentially $H_1$-graphic or the complementary sequence $\overline{\pi}=(N-1-d_N,\dots, N-1-d_1)$ is potentially $H_2$-graphic. We prove that if $s\ge 2$ is an integer and $T_t$ is a tree of order $t> 7(s-2)$, then $$r_{pot}(K_s, T_t) = t+s-2.$$ This result, which is best possible up to the bound on $t$, is a degree sequence analogue to a classical 1977 result of Chv\'{a}tal on the graph Ramsey number of trees vs. cliques. To obtain this theorem, we prove a sharp condition that ensures an arbitrary graph packs with a forest, which is likely to be of independent interest.