度数和主循环的极值图

IF 0.6 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society Pub Date : 2024-09-13 DOI:10.1017/s0004972724000522

LU CHEN, YUEYU WU

引用次数: 0

摘要

如果 $V(C)$ 是支配集且 $V(G)\backslash V(C)$ 是独立集，那么图 G 的循环 C 就是支配集。Wu 等人['度和与支配循环'，《离散数学》344 (2021)，文章编号：112224]证明了每一个最长的循环都是支配循环。112224]证明了如果对于任意 $k+1$ 成对非相邻顶点的度数总和大于 $(k+1)(n+1)/3$，则在 $n\geq 3$ 顶点上具有 $k\geq 2$ 的 k 连接图 G 的每个最长循环都是支配循环。他们还证明了这一界限是尖锐的。在本文中，我们证明了这个条件下的极值图 G 满足 $(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ 或 $2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$ 。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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EXTREMAL GRAPHS FOR DEGREE SUMS AND DOMINATING CYCLES

A cycle C of a graph G is dominating if $V(C)$ is a dominating set and $V(G)\backslash V(C)$ is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on $n\geq 3$ vertices with $k\geq 2$ is dominating if the degree sum is more than $(k+1)(n+1)/3$ for any $k+1$ pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy $(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ or $2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$

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来源期刊

Bulletin of the Australian Mathematical Society 数学-数学

CiteScore

1.20

自引率

14.30%

发文量

149

审稿时长

4-8 weeks

期刊介绍： Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way. Published Bi-monthly Published for the Australian Mathematical Society