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Coloring graphs with no long induced path — arXiv:2609.08847

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08847
License (as archived): CC0-1.0
Archived: 2026-10-16T14:49:06.519Z
SHA-256 of the archived text: 822c4eebefbc576a0ea4f947b15d5844d4a9eb7896809536e65fa77a3d54a386
Stable URL of this page: https://metraipolis.com/documento/dc_850db2498346

How to cite

Coloring graphs with no long induced path — arXiv:2609.08847 — arXiv (en), https://arxiv.org/abs/2609.08847 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_850db2498346 (Metraipolis archive dc_850db2498346, sha256 822c4eebefbc576a0ea4f947b15d5844d4a9eb7896809536e65fa77a3d54a386)

BibTeX

@misc{dc850db2498346,
  title = {Coloring graphs with no long induced path — arXiv:2609.08847},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08847},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_850db2498346; sha256 822c4eebefbc576a0ea4f947b15d5844d4a9eb7896809536e65fa77a3d54a386},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08847. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)-1}$ for $t\ge4$. We lower the base of the exponential by one: for every $t\ge5$, every $P_t$-free graph $G$ satisfies \[ \chi(G)\le 3\,(t-3)^{\omega(G)+4}. \] The proof combines two refinements of the Gyárfás path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Discrete Mathematics (cs.DM).
Pagina del lavoro: https://arxiv.org/abs/2609.08847 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:32:29.882Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).