Isoperimetry, Stability, and Irredundance in Direct Products

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 an optimal vertex isoperimetric inequality for direct products of complete multipartite graphs. Applying this inequality, we derive a stability result for independent sets in direct products of balanced complete multipartite graphs, showing that every large independent set must be close to the maximal independent set determined by setting one of the coordinates to be constant. Armed with these isoperimetry and stability results, we prove that the upper irredundance number of a direct product of balanced complete multipartite graphs is equal to its independence number in all but at most $37$ cases. This proves most of a conjecture of Burcroff that arose as a strengthening of a conjecture of the second author and Iyer. We also propose a further strengthening of Burcroff's conjecture.Comment: 24 pages, 0 figure

Tightness of Paired and Upper Domination Inequalities for Direct Product Graphs

A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $\gamma_{\mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $\Gamma(G)$ is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of two multiplicative inequalities for these domination parameters, where the graph product is the direct product $\times$. We show that for every positive constant $c$, there exist graphs $G$ and $H$ of arbitrarily large diameter such that $\gamma_{\mathrm{pr}}(G \times H) \leq c\gamma_{\mathrm{pr}}(G)\gamma_{\mathrm{pr}}(H)$, thus answering a question of Rall as well as two questions of Paulraja and Sampath Kumar. We then study when this inequality holds with $c = \frac{1}{2}$, in particular proving that it holds whenever $G$ and $H$ are trees. Finally, we demonstrate that the inequality $\Gamma(G \times H) \geq \Gamma(G) \Gamma(H)$, due to Bre\v{s}ar, Klav\v{z}ar, and Rall, is tight.Comment: 15 pages, 3 figure

Sets Arising as Minimal Additive Complements in the Integers

A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that arise as minimal additive complements to finite sets. We provide some necessary conditions for an eventually periodic set to arise as a minimal additive complement and demonstrate that these necessary conditions are also sufficient for certain classes of eventually periodic sets. We conclude with several conjectures and questions concerning the structure of minimal additive complements.Comment: 15 page