On generalised Petersen graphs of girth 7 that have cop number 4

Harmony Morris, Joy Morris
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引用次数: 0

Abstract

We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most $4$. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number $4$ but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number $4$ and girth $8$, while this paper explains those that have girth $7$.)
周长为7的广义Petersen图上的系数为4
我们证明,如果$n=7k/i$且$i \in \{1,2,3\}$,则广义Petersen图$GP(n,k)$的cop数为$4$,除了一些先前已知的小例外。Ball et al.(2015)先前证明了任何广义Petersen图的cop数最多为$4$。本文的结果解释了所有已知的广义Petersen图,这些图实际上具有cop数$4$,但Morris等人在最近的预印本中没有解释过,并将它们置于无限族的背景下。(更准确地说,Morris等人的预印本解释了所有已知的周长为8美元的广义Petersen图,而本文解释了周长为7美元的广义Petersen图。)
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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