Sharp results for the Erdős, Pach, Pollack and Tuza problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912230570721280 |
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| author | Cambie, Stijn Jooken, Jorik |
| author_facet | Cambie, Stijn Jooken, Jorik |
| contents | We consider the Erdős, Pach, Pollack and Tuza problem, asking for the maximum diameter of a graph with given order $n$, minimum degree $δ$ and clique number at most $ω$. We solve their problem asymptotically for the first hard case, $ω\leq 3$, for the smallest values of $δ$ by determining the smallest rational number $f(δ)$ such that $diam(G) \leq f(δ)n+O(1)$ for all graphs $G$ with order $n$, minimum degree $δ$ and clique number $ω\leq 3$. We also consider the weaker version where the clique number $ω\leq 3$ is replaced by having chromatic number $χ\leq 3$ and solve this version for small $δ$, thereby yielding a counterexample to a conjecture of Erdős et al. in a regime where this conjecture was still open. When restricting the conjecture to graphs with chromatic number $χ\leq 3$, we show that this counterexample appears for the smallest possible $δ$, namely $δ=16.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp results for the Erdős, Pach, Pollack and Tuza problem Cambie, Stijn Jooken, Jorik Combinatorics 05C35, 05C12, 05C07 We consider the Erdős, Pach, Pollack and Tuza problem, asking for the maximum diameter of a graph with given order $n$, minimum degree $δ$ and clique number at most $ω$. We solve their problem asymptotically for the first hard case, $ω\leq 3$, for the smallest values of $δ$ by determining the smallest rational number $f(δ)$ such that $diam(G) \leq f(δ)n+O(1)$ for all graphs $G$ with order $n$, minimum degree $δ$ and clique number $ω\leq 3$. We also consider the weaker version where the clique number $ω\leq 3$ is replaced by having chromatic number $χ\leq 3$ and solve this version for small $δ$, thereby yielding a counterexample to a conjecture of Erdős et al. in a regime where this conjecture was still open. When restricting the conjecture to graphs with chromatic number $χ\leq 3$, we show that this counterexample appears for the smallest possible $δ$, namely $δ=16.$ |
| title | Sharp results for the Erdős, Pach, Pollack and Tuza problem |
| topic | Combinatorics 05C35, 05C12, 05C07 |
| url | https://arxiv.org/abs/2502.08626 |