Graph Sensitivity of Cartesian Products with Matched Bridges

Abstract

For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha (G)+t$ vertices, where $\alpha (G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge \lceil \sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to Cartesian products of $k$ bipartite graphs with perfect matchings and prove that equality holds when the factors are connected and each has a matched bridge. In particular, we prove that $f(T_1\square \cdots \square T_k)=\lceil \sqrt{k}\rceil$ whenever each $T_i$ is a tree with a perfect matching. We determine the sensitivity of every Cartesian product of paths, settling the even-path case left open by Zeng and Hou [J. Graph Theory 107 (2024), 169--180]. For these tree products, with $D=\lceil \sqrt{k}\rceil$, we also prove that $f_t(T_1\square \cdots \square T_k)=D$ whenever $1\le t\le 2^{D-\lceil {\log }_{2}D\rceil -1}$. When $t=2$, this equality holds for all $k\ge 2$, provided that at least one factor is not $K_2$. Matching cuts give an additional exact range for products of even-order paths. Finally, we prove that $f_2(Q_k)=\lceil \sqrt{k}\rceil$ for every $k\ge 2$ with $k\notin \{4,9\}$, whereas $f_2(Q_4)=3$ and $3\le f_2(Q_9)\le 4$.

17 pages