Graphs and Matroids - Aristotelis Chaniotis

Title: Induced subgraphs of graphs of large $K_{r}$-free chromatic number

Abstract:For an integer $r\geq 2$, the $K_{r}$-free chromatic number of a graph $G$, denoted by $\chi_{r}(G)$, is the minimum size of a partition of the set of vertices of $G$ into parts each of which induces a $K_{r}$-free graph. In this setting, the $K_{2}$-free chromatic number is the usual chromatic number. Which are the unavoidable induced subgraphs of graphs of large $K_{r}$-free chromatic number? Generalizing the notion of $\chi$-boundedness, we say that a hereditary class of graphs is $\chi_{r}$-bounded if there exists a function which provides an upper bound for the $K_{r}$-free chromatic number of each graph of the class in terms of the graph's clique number. With an emphasis on a generalization of the Gy\'arf\'as-Sumner conjecture for $\chi_{r}$-bounded classes of graphs and on polynomial $\chi$-boundedness, I will discuss some recent developments on $\chi_{r}$-boundedness and related open problems. Based on joint work with Mathieu Rundstr\"om and Sophie Spirkl.