# 清华大学陆玫教授；山东大学吴建良教授；南京大学陈耀俊教授 学术报告

Theorem: Let G be a $K_5$-minor free graph.

(1) If $\Delta(G)\geq 7$,then $\chi'(G) =\Delta(G)$;
(2)$\lceil\frac{\Delta(G)}{2}\rceil\leq la(G) \leq\lceil\frac{\Delta(G)+1}{2}\rceil$. Moreover, if $\Delta(G)\geq 9$, then $la(G)= \lceil\frac{\Delta(G)}{2}\rceil$;

(3) If $\Delta(G)\geq 7$, then $\chi''(G) =\Delta(G)+2$. Moreover, if $\Delta(G)\geq 10$, then $\chi''(G)=\Delta(G)+1$.

摘要：For two given sets C1 and C2 of cycles, the Ramsey number R(C1,C2) is the smallest integer N such that for any graph G on N vertices, either G contains a cycle from C1 or its complement contains a cycle from C2. In this paper, we determine all  Ramsey numbers R(C1,C2), which confirms a conjecture due to Hansson recently, and extends the well known Ramsey numbers for two cycles.