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The direct product of graphs $G_1,\ldots,G_n$ is the graph with vertex set$V(G_1)\times\cdots\times V(G_n)$ in which two vertices $(g_1,\ldots,g_n)$ and$(g_1',\ldots,g_n')$ are adjacent if and only if $g_i$ is adjacent to $g_i'$ in$G_i$ for all $i$. Building off of the recent work of Brakensiek, we prove anoptimal vertex isoperimetric inequality for direct products of completemultipartite graphs. Applying this inequality, we derive a stability result forindependent sets in direct products of balanced complete multipartite graphs,showing that every large independent set must be close to the maximalindependent set determined by setting one of the coordinates to be constant.Armed with these isoperimetry and stability results, we prove that the upperirredundance number of a direct product of balanced complete multipartitegraphs is equal to its independence number in all but at most $37$ cases. Thisproves most of a conjecture of Burcroff that arose as a strengthening of aconjecture of the second author and Iyer. We also propose a furtherstrengthening of Burcroff's conjecture.

Isoperimetry, Stability, and Irredundance in Direct Products