发表机构
Seoul National University; Korea Institute for Advanced Study (KIAS)（首尔大学; 韩国高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明连通三次图是det-极值的当且仅当其全支配数为1，推广了McCuaig的刻画，并给出非二分情形的最小顶点数界及围长性质。
AI 中文摘要
一个图 $G$ 称为 det-极值的，如果其邻接矩阵 $A$ 满足 $|\operatorname{det} A|=\operatorname{per} A$。det-极值三次二分图出现在 Pólya 的永久性问题研究中，McCuaig 将 3-连通的此类图刻画为 Heawood 图的顶点和。图的全支配数是两两不相交的全支配集的最大数目。具有全支配数 $1$ 的三次图的刻画一直是一个长期未解决的问题。在本文中，我们证明一个连通三次图是 det-极值的当且仅当其全支配数为 $1$。我们进一步证明 McCuaig 的刻画可以推广到所有 3-连通三次图，并且每个连通的 det-极值三次图具有围长 $3$、$5$ 或 $6$。我们还证明了一个连通的 det-极值三次非二分图至少有 $28$ 个顶点，并且这个界是最优的。通过这种对应关系，这些结果可以推广到全支配数为 $1$ 的三次图。此外，用构形的语言来说，我们的结果意味着每个无三角形 $3$-构形都有一个阻塞集。
英文摘要
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.
Comments24 pages, 5 figures
