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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.21127 |
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Table of Contents:
- Let $T$ be a forest. We study polynomially high-chromatic pure pairs in graphs with no $T$ as an induced subgraph ($T$-free graphs in other words), with applications to the polynomial Gyárfás-Sumner conjecture. In addition to reproving several known results in the literature, we deduce: $\bullet$ If $T=P_5$ is the five-vertex path, then every $T$-free graph $G$ with clique number $w\ge2$ contains a complete pair $(A,B)$ of induced subgraphs with $χ(A)\ge w^{-d}χ(G)$ and $χ(B)\ge 2^{-d}χ(G)$, for some universal $d\ge1$. The proof uses the recent Erdős-Hajnal result for $P_5$-free graphs. Via the classical Gyárfás path argument, such a ``polynomial versus linear high-$χ$ complete pairs'' result can be viewed as further supporting evidence for the polynomial Gyárfás-Sumner conjecture for $P_5$. In particular, it implies \[χ(G)\le w^{O(\log w/\log\log w)}\] which asymptotically improves the bound $χ(G)\le w^{\log w}$ of Scott, Seymour, and Spirkl. $\bullet$ If $T$ and a broom satisfy the polynomial Gyárfás-Sumner conjecture, then so does their disjoint union. Unifying earlier results of Chudnovsky, Scott, Seymour, and Spirkl, and of Scott, Seymour, and Spirkl, this gives new instances of $T$ for which the conjecture holds.