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

Shenwei Huang, Yidong Zhou, Yeonsu Chang

TL;DR

This work proves that every ($P_7$, even-hole)-free graph $G$ satisfies $\chi(G)\le \left\lceil \frac{5}{4}\omega(G)\right\rceil$, with the bound attained by equal-size blowups of $C_5$, establishing optimal χ-boundedness for this class. The authors develop a heavy structural framework centered on nice blowups of $C_5$ and a detailed attachment decomposition into $A_0,A_1,A_2,A_3,A_5$, complemented by clique-cutset arguments, $(p,q)$-good subgraphs, and a descent-type strategy to color the graph. A key technical lemma and a sequence of reduction lemmas enable construction of $(4,5)$-good subgraphs and precolored cores that cap $\chi(G)$ by the target bound, while also confirming Reed's Conjecture for this graph class. The results advance the understanding of χ-boundedness for even-hole-free graphs and connect to conjectures on all $5$-hole graphs, offering a path toward a tighter global bound for this important graph family.

Abstract

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).

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

TL;DR

This work proves that every (, even-hole)-free graph satisfies , with the bound attained by equal-size blowups of , establishing optimal χ-boundedness for this class. The authors develop a heavy structural framework centered on nice blowups of and a detailed attachment decomposition into , complemented by clique-cutset arguments, -good subgraphs, and a descent-type strategy to color the graph. A key technical lemma and a sequence of reduction lemmas enable construction of -good subgraphs and precolored cores that cap by the target bound, while also confirming Reed's Conjecture for this graph class. The results advance the understanding of χ-boundedness for even-hole-free graphs and connect to conjectures on all -hole graphs, offering a path toward a tighter global bound for this important graph family.

Abstract

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 -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 without induced 7-vertex paths satisfies , where and denote the chromatic number and clique number of , 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).
Paper Structure (29 sections, 178 theorems, 27 equations, 9 figures)

This paper contains 29 sections, 178 theorems, 27 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Containing an induced $C_7$
  4. Nice blowups of $C_5$
  5. Basic properties
  6. Anticomplete pairs
  7. $A_0$ is empty
  8. Neighborhoods of $A_1(i)$
  9. Neighborhoods of $A_2(i)$
  10. Neighborhoods of $A_3(i)$
  11. Single $A_3(i)$
  12. Multiple $A_3(i)$
  13. Small vertex argument for $A_3(i)$
  14. Non-neighbors of $C_5$
  15. A key technical lemma
  16. ...and 14 more sections

Key Result

Theorem 1.1

For every ($P_7$, even-hole)-free graph $G$, $\chi(G)\le \left\lceil \frac{5}{4}\omega(G) \right\rceil$.

Figures (9)

  • Figure 1: Three special graphs $M$, $M_1$ and $M_2$.
  • Figure 2: A $(4,5)$-good subgraph in Lemma \ref{['lem:general 2']}.
  • Figure 3: A $(4,5)$-good subgraph in Lemma \ref{['lem:L_8 L_9 L_10 are not empty']}.
  • Figure 4: $(4,5)$-good subgraphs in Lemmas \ref{['lem:exactly two S_5']}-\ref{['lem:only one S_Y']}.
  • Figure 5: A basic graph arising from two non-empty $A_1(j)$s. A double line means complete.
  • ...and 4 more figures

Theorems & Definitions (356)

  • Theorem 1.1
  • Lemma 2.1: NS01
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Dirac61
  • Definition 2.5: Minimal simplicial vertices
  • Lemma 2.6: Two non-adjacent minimal simplicail vertices
  • proof
  • ...and 346 more