The optimal chromatic bound for even-hole-free graphs without induced seven-vertex paths
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2026
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| author | Huang, Shenwei Zhou, Yidong Chang, Yeonsu |
| author_facet | Huang, Shenwei Zhou, Yidong Chang, Yeonsu |
| contents | The class of even-hole-free graphs has been extensively studied on its own and on its relation to perfect graphs. In this paper, we study the $χ$-boundedness of even-hole-free graphs which itself is an important topic in graph theory. In particular, we prove that every even-hole-free graph $G$ without induced 7-vertex paths satisfies $χ(G)\le \lceil\frac{5}{4}ω(G)\rceil$, where $χ(G)$ and $ω(G)$ denote the chromatic number and clique number of $G$, respectively. This bound is optimal. Our result strictly extends the result of Karthick and Maffary \cite{KM19} on even-hole-free graphs without induced 6-vertex paths, and implies that even-hole-free graphs without induced 7-vertex paths satisfy Reed's Conjecture. Our proof relies on a heavy structural analysis on a maximal substructure called a nice blowup of a five-cycle and can be viewed for graphs in which all holes are of length five (graphs with all holes having the same length gain increasing interest in recent years \cite{COOK202496}). Our result gives a partial answer to a conjecture of Wang and Wu \cite{WW25} on graphs in which all holes are of length 5. One of the key technical ingredients is a technical lemma proved via clique cutset argument combined with the idea of Infinite Descent Method (often used in number theory). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04403 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The optimal chromatic bound for even-hole-free graphs without induced seven-vertex paths Huang, Shenwei Zhou, Yidong Chang, Yeonsu Combinatorics The class of even-hole-free graphs has been extensively studied on its own and on its relation to perfect graphs. In this paper, we study the $χ$-boundedness of even-hole-free graphs which itself is an important topic in graph theory. In particular, we prove that every even-hole-free graph $G$ without induced 7-vertex paths satisfies $χ(G)\le \lceil\frac{5}{4}ω(G)\rceil$, where $χ(G)$ and $ω(G)$ denote the chromatic number and clique number of $G$, respectively. This bound is optimal. Our result strictly extends the result of Karthick and Maffary \cite{KM19} on even-hole-free graphs without induced 6-vertex paths, and implies that even-hole-free graphs without induced 7-vertex paths satisfy Reed's Conjecture. Our proof relies on a heavy structural analysis on a maximal substructure called a nice blowup of a five-cycle and can be viewed for graphs in which all holes are of length five (graphs with all holes having the same length gain increasing interest in recent years \cite{COOK202496}). Our result gives a partial answer to a conjecture of Wang and Wu \cite{WW25} on graphs in which all holes are of length 5. One of the key technical ingredients is a technical lemma proved via clique cutset argument combined with the idea of Infinite Descent Method (often used in number theory). |
| title | The optimal chromatic bound for even-hole-free graphs without induced seven-vertex paths |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.04403 |