Det-extremal cubic graphs and the total domatic number

Abstract

A graph $G$ is det-extremal if $|\operatorname{det} A|=\operatorname{per} A$ for its adjacency matrix $A$. Det-extremal cubic bipartite graphs arise in the study of Pólya's permanent problem, and McCuaig characterized the $3$-connected ones as vertex-sums of copies of the Heawood graph. The total domatic number of a graph is the largest number of pairwise disjoint total dominating sets. Characterization of the cubic graphs with total domatic number $1$ has been a long-standing open problem. In this paper, we prove that a connected cubic graph is det-extremal if and only if its total domatic number is $1$. We further show that McCuaig's characterization extends to all $3$-connected cubic graphs, and that every connected det-extremal cubic graph has girth $3$, $5$ or $6$. We also prove that a connected det-extremal cubic non-bipartite graph has at least $28$ vertices, and that this bound is best possible. Through this correspondence, these results carry over to cubic graphs with total domatic number $1$. In addition, in the language of configurations, our results imply that every triangle-free $3$-configuration has a blocking set.