This week's seminar is given by our own Altar Ciceksiz, Umeå universitet.
Title: Connectivity threshold for the Square Graph
Abstract: Given a random graph G chosen from the Erdős-Rényi random graph on n vertices with edge probabilities p, we consider the percolation properties of the associated square graph S(G). The square graph is the graph with the vertices V(S(G)):={S=xyzt: xyzt is a 4 cycle in G}, and edges E(S(G)):={ST: the induced 4 cycles S and T share a diagonal}. In 2021, Falgas-Ravry, Behrstock, Hagen and Susse determined the critical probability for the threshold of a giant component emerging in the square graph, and in 2021 Susse conjectured that the threshold for the square graph S(G) being connected. We settle the conjecture of Susse. We will also briefly talk about the geometric group theory context and what it means to associate a group to a random graph.
