Cops and Robbers on Graphs with Path Constraints
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arXiv
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| Format: | Preprint |
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2025
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| author | Clow, Alexander Meger, Erin |
| author_facet | Clow, Alexander Meger, Erin |
| contents | In 2019, Sivaraman conjectured that every $P_k$-free graph has cop number at most $k-3$. In the same year, Liu proved this conjecture for $(P_k,\text{claw})$-free graphs. Recently Chudnovsky, Norin, Seymour, and Turcotte proved this conjecture for $P_5$-free graphs. For $k\geq 6$ the conjecture remains widely opened. Let the $E$ graph be the $\text{claw}$ with two subdivided edges. We show that all $(P_k,E)$-free graphs have cop number at most $\lceil \frac{k-1}{2} \rceil +3$, which improves and generalizes Liu's result for $(P_k,\text{claw})$-free graphs. We also prove that if $G$ is a graph whose longest path is length $p$, then $G$ has cop number at most $\lceil \frac{2p}{3} \rceil+3$. This improves a bound of Joret, Kamiński, and Theis. Our proof relies on demonstrating that all $(P_k,\text{claw},\text{butterfly},C_4,C_5)$-free graphs have cop number at most $\lceil\frac{k-1}{3}\rceil +3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cops and Robbers on Graphs with Path Constraints Clow, Alexander Meger, Erin Combinatorics 05C57 In 2019, Sivaraman conjectured that every $P_k$-free graph has cop number at most $k-3$. In the same year, Liu proved this conjecture for $(P_k,\text{claw})$-free graphs. Recently Chudnovsky, Norin, Seymour, and Turcotte proved this conjecture for $P_5$-free graphs. For $k\geq 6$ the conjecture remains widely opened. Let the $E$ graph be the $\text{claw}$ with two subdivided edges. We show that all $(P_k,E)$-free graphs have cop number at most $\lceil \frac{k-1}{2} \rceil +3$, which improves and generalizes Liu's result for $(P_k,\text{claw})$-free graphs. We also prove that if $G$ is a graph whose longest path is length $p$, then $G$ has cop number at most $\lceil \frac{2p}{3} \rceil+3$. This improves a bound of Joret, Kamiński, and Theis. Our proof relies on demonstrating that all $(P_k,\text{claw},\text{butterfly},C_4,C_5)$-free graphs have cop number at most $\lceil\frac{k-1}{3}\rceil +3$. |
| title | Cops and Robbers on Graphs with Path Constraints |
| topic | Combinatorics 05C57 |
| url | https://arxiv.org/abs/2509.10941 |