Michał Pilipczuk, Structural properties of powers of sparse graphs

For a graph G and an integer d, the dth power of G is the graph $G^d$ on the same vertex set as G where two vertices are considered adjacent if and only if they are at distance at most d in G. Assuming that G is sparse, what can we say about the structure in $G^d$? Certainly $G^d$ can be dense, as the square of a star is a complete graph, but $G^d$ still retains many properties that can be derived from the sparsity of G. We will present some recent results in this spirit, in particular connected to colorings and to the Erdős-Hajnal property, assuming that G is drawn from a fixed class of bounded expansion or from a fixed nowhere dense class. The talk will be based on the recent papers: “Clustering Powers of Sparse Graphs” (with J. Nešetřil, P. Ossona de Mendez, and X. Zhu) and “Erdős-Hajnal properties for powers of sparse graphs” (with M. Briański, P. Micek, and M. Seweryn).