Cops and Robbers, Clique Covers, and Induced Cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915400513486848 |
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| author | Clow, Alexander Zaguia, Imed |
| author_facet | Clow, Alexander Zaguia, Imed |
| contents | We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq α(G) \leq θ(G)$ where $α(G)$ and $θ(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < α(G)$ for all graphs with $α(G) \geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $α= 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $k\geq 1$ there exists a graph $G$ such that $c(G) = α(G) = θ(G) = k$. Next, we consider the structure of graphs with $c(G) = θ(G) \geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = θ(G) = k \geq 3$ contain induced cycles of all lengths $3\leq t \leq k+1$. This implies all perfect graphs $G$ with $α(G)\geq 4$ have $c(G) < α(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < α(G)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cops and Robbers, Clique Covers, and Induced Cycles Clow, Alexander Zaguia, Imed Combinatorics Discrete Mathematics 05C57, 05C80, 05C17 We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq α(G) \leq θ(G)$ where $α(G)$ and $θ(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < α(G)$ for all graphs with $α(G) \geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $α= 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $k\geq 1$ there exists a graph $G$ such that $c(G) = α(G) = θ(G) = k$. Next, we consider the structure of graphs with $c(G) = θ(G) \geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = θ(G) = k \geq 3$ contain induced cycles of all lengths $3\leq t \leq k+1$. This implies all perfect graphs $G$ with $α(G)\geq 4$ have $c(G) < α(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < α(G)$. |
| title | Cops and Robbers, Clique Covers, and Induced Cycles |
| topic | Combinatorics Discrete Mathematics 05C57, 05C80, 05C17 |
| url | https://arxiv.org/abs/2507.14321 |