发表机构
East China Normal University; Jiangsu Normal University（华东师范大学; 江苏师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了Desormeaux和Henning关于三次二分图配对支配数的猜想，通过多种技术结合给出尖锐上界，并确定等号情形。
AI 中文摘要
图$G$的配对支配集是满足$G[D]$具有完美匹配的支配集$D$。此类集合的最小规模称为配对支配数$\gpr(G)$。Desormeaux和Henning猜想：每个阶为$n$的三次二分图$G$满足$\gpr(G)\le n/2$。我们以尖锐整数形式证明了该猜想，即对于每个有限简单三次二分图$G$，有$\gpr(G)\le 2\lfloor |V(G)|/4\rfloor$。证明结合了沿完美匹配的有向收缩、基于支配树的切换论证、用于二边割的四符号边界演算以及Gallai--Edmonds分解。当$|V(G)|\equiv2\pmod4$时，等号由$K_{3,3}$取得；当$|V(G)|\equiv0\pmod4$时，等号由立方体$Q_3$取得。
英文摘要
A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in the sharp integer form $\gpr(G)\le 2\lfloor |V(G)|/4\rfloor$ for every finite simple cubic bipartite graph $G$. The proof combines a directed contraction along a perfect matching, switching arguments based on dominator trees, a four-symbol boundary calculus for two-edge cuts, and the Gallai--Edmonds decomposition. Equality is attained by $K_{3,3}$ when $|V(G)|\equiv2\pmod4$ and by the cube $Q_3$ when $|V(G)|\equiv0\pmod4$.
