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Meyniel's conjecture on graphs of bounded degree
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2021-03-01 , DOI: 10.1002/jgt.22662
Seyyed Aliasghar Hosseini 1 , Bojan Mohar 1 , Sebastian Gonzalez Hermosillo de la Maza 1
Affiliation  

The game of Cops and Robbers is a well‐known pursuit‐evasion game played on graphs. It has been proved that cubic graphs can have arbitrarily large cop number c ( G ) , but the known constructions show only that the set { c ( G ) G cubic } is unbounded. In this paper, we prove that there are arbitrarily large subcubic graphs G whose cop number is at least n 1 2 o ( 1 ) where n = V ( G ) . We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weaker version of Meyniel's conjecture for all graphs.

中文翻译:

有限度图上的梅尼尔猜想

警察与强盗的游戏是一种著名的追逃游戏,使用图表进行游戏。已经证明三次图可以具有任意大的cop数 C G ,但已知的构造仅显示该集合 { C G G 立方体 } 是无界的。在本文中,我们证明存在任意大的亚三次图 G 警察号码至少为 ñ 1个 2个 - Ø 1个 在哪里 ñ = 伏特 G 。我们还表明,证明有界图的Meyniel猜想意味着对所有图的Meyniel猜想的较弱版本。
更新日期:2021-03-01
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