Ringi Kim (김린기), The strong clique number of graphs with forbidden cycles

The strong clique number of a graph $G$ is the maximum size of a set of edges of which every pair has distance at most two.

In this talk, we prove that every  $\{C_5,C_{2k}\}$-free graph has strong clique number at most $k\mathrm{\Delta }(G)-(k-1)$, which resolves a conjecture by  Cames van Batenburg et al. We also prove that every $C_{2k}$-free graph has strong clique number at most $(2k-1)\mathrm{\Delta }(G)+(2k-1{)}^2$, improving the previous known upper bound $10k^2(\mathrm{\Delta }(G)-1)$ due to  Cames van Batenburg et al. This is joint work with Eun-Kyung Cho, Ilkyoo Choi, and Boram Park.