Cooling Generalized Hypercubes

Teddy Mishura, Toronto Metropolitan University

Graph cooling was introduced as an analogue to the well known game of graph burning. Both of these games are round based processes where fires spread from burning vertices to adjacent vertices and the player (the Arsonist) lights a vertex on fire every round. In graph burning, the player must burn all the vertices of G as quickly as possible, whereas in cooling, the player must burn all the vertices of G as slowly as possible. The cooling number of the hypercube graph Qn, written CL(Qn), was recently shown to be exactly n.

In this talk, I analyze the cooling number of two families of graphs that can be viewed as the extension of the hypercube graph Qn — Cartesian products of complete graphs Km and Cartesian products of path graphs Pk. In the former case, I present an explicit form for the cooling number, and in the latter case, I present asymptotically tight bounds. I end the talk with a brief discussion on open problems.

Based on joint work with Anthony Bonato, MacKenzie Carr, Caleb Jones, and Trent Marbach.