The optimal chromatic bound for even-hole-free graphs without induced seven-vertex paths

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).

💡 Research Summary

The paper investigates the chromatic number (χ(G)) of even‑hole‑free graphs that also forbid an induced path on seven vertices ((P_7)). An even‑hole‑free graph contains no induced cycles of even length, a class that has been intensively studied because of its close relationship with perfect graphs. While it is known that every even‑hole‑free graph admits a linear χ‑bounding function (Chudnovsky–Seymour proved that such graphs have a vertex whose neighbourhood can be covered by two cliques, yielding (χ(G) ≤ 2ω(G)−1)), the exact optimal bound has remained open.

The main result of the paper (Theorem 1.1) is that for every graph (G) that is both (P_7)‑free and even‑hole‑free, \