Seminar: Liminal burning the hypercube

Speaker: Theodore Mishura, TMU

Title: Liminal burning the hypercube

Abstract: Liminal burning generalizes both the burning and cooling processes in graphs. In $k$-liminal burning, a Saboteur reveals $k$-sets of vertices in each round, and the Arsonist must choose sources only within these sets. The result is a two-player game with the corresponding optimization parameter $b_k$ called the $k$-liminal burning number. For $k = |V (G)|$, liminal burning is identical to burning, and for $k = 1$, liminal burning is identical to cooling. Here, we study the behavior of $k$-liminal burning on the hypercube graph $Q_n$ and note that finding the $k$-liminal burning number of $Q_n$ is strongly related to finding an appropriate Sperner family—a family of sets where no element is a proper subset of another. We introduce a variant of these Sperner families that, alongside other methods, allows us to establish bounds on $b_k(Q_n)$ for various values of $k$. We also determine the exact cooling number of the $n$-dimensional hypercube to be $n.$

Joint work with: Anthony Bonato, Trent Marbach, John Marcoux.