矩形网格图中的上支配数及其与子法定人数染色的关系

中文摘要 AI 辅助

对于图 $G$，上支配数 $\Gamma(G)$ 是最小支配集的最大基数。2023 年的一项计算研究将方形网格图的上支配数列入尚无通用公式的情形。我们观察到，对于二部图，经典等式 $\Gamma(G)=\alpha(G)$ 立即确定了每个矩形网格图 $G_{m,n}=P_m\square P_n$ 的参数。即，\\[ \Gamma(G_{m,n})=\left\lceil\frac{mn}{2}\right\rceil. \\] 我们给出两个简短证明，一个通过经典二部上支配定理，另一个通过匹配界 $\Gamma(G)+\alpha'(G)\le |V(G)|$。然后我们将上支配数与网格图的 $2$-独立数和子法定人数染色数进行比较。这产生了无条件链 \\[ \Gamma(G_{m,n})\le \beta_2(G_{m,n})\le \psi_{\mathrm{sq}}(G_{m,n}), \\] 以及 $\Gamma(G_{m,n})$ 与 $\beta_2(G_{m,n})$ 之间间隙的显式公式。

英文摘要

For a positive integer $k$, let $β_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $β_k(G)\le SF_k(G)$ suggests asking when equality holds. We place this question in the framework of upper domination: at $k=1$, $β_1(G)=α(G)$ and $SF_1(G)=Γ(G)$, so the classical equality $Γ=α$ on bipartite graphs is exactly the first member of the proposed hierarchy. We prove the equality for complete bipartite graphs for every $k$, obtaining \[ β_k(K_{a,b})=SF_k(K_{a,b})=\max\{a,b,2k-2\}\qquad(a,b\ge k), \] and record the elementary low-degree case $Δ(G)<k$. For $k=2$ we derive certificate restrictions for minimal $2$-star-forming sets in bipartite graphs. For chain graphs we go further: we prove $β_2(G)=SF_2(G)$ and obtain an exact formula for their common value. The proof uses the nested-neighborhood structure together with a classification of witnesses to the indispensability of a high internal-degree vertex. We retain the equality problem for chain graphs as a conjecture only for $k\ge3$, and formulate the broader bipartite conjecture. We also determine the upper domination number of every rectangular grid and combine it with the known exact dissociation number to compare $Γ$, $β_2$, and $SF_2$. In particular, $Γ=β_2$ on every even-by-even rectangular grid, while $β_2\le SF_2$ always; this motivates a grid equality conjecture whose even-by-even case would yield a three-parameter identity.

发表机构

• National Higher School of Advanced Technologies（国家高等先进技术学院）

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