Ramsey-type $χ$-bounds for $χ$-bounded graph classes
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arXiv
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| Format: | Preprint |
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2026
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| author | Nguyen, Tung Oum, Sang-il |
| author_facet | Nguyen, Tung Oum, Sang-il |
| contents | We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $χ$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(α(H),ω(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above:
(1) every component of $T$ is a broom and $H$ is complete multipartite; or
(2) $T$ is a forest and $H$ is complete bipartite.
These two unify and substantially extend a number of previous results on linear and polynomial $χ$-boundedness for various graph classes.
For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08848 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramsey-type $χ$-bounds for $χ$-bounded graph classes Nguyen, Tung Oum, Sang-il Combinatorics 05C15, 05C35, 05C55, 05C69, 05C75 We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $χ$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(α(H),ω(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $χ$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions. |
| title | Ramsey-type $χ$-bounds for $χ$-bounded graph classes |
| topic | Combinatorics 05C15, 05C35, 05C55, 05C69, 05C75 |
| url | https://arxiv.org/abs/2605.08848 |