Induced subgraphs and tree decompositions XI. Local structure in even-hole-free graphs of large treewidth

TL;DR

This paper proves a conjecture that large even-hole-free graphs contain specific four-vertex subgraphs, and extends the result to include larger complete subgraphs and certain induced subgraphs related to chordal graphs.

Contribution

It establishes new structural results about even-hole-free graphs with large treewidth, confirming the conjecture for specific subgraph configurations and extending to broader classes of induced subgraphs.

Findings

Large even-hole-free graphs contain a four-vertex subgraph with at least five edges.

The results are tight for the classes of graphs considered, with counterexamples outside these classes.

The paper generalizes the conjecture to include larger complete subgraphs and certain induced subgraphs.

Abstract

We prove a conjecture of Sintiari and Trotignon that every even-hole-free graph of sufficiently large treewidth contains a four-vertex induced subgraph with at least five edges (that is, either the four-vertex complete graph or the unique four-vertex graph with five edges, also known as the diamond). In fact, we prove two stronger results: (a) For every $K_4$-free chordal graph $H$, every even-hole-free graph of sufficiently large treewidth contains either a four-vertex complete subgraph or an induced subgraph isomorphic to $H$ (when $H$ is the diamond, this yields their conjecture); and (b) For every $K_3$-free chordal graph $H$ (equivalently, for every forest $H$) and every $t \in \mathbb{N}$, every even-hole-free graph of sufficiently large treewidth contains either a $t$-vertex complete subgraph or an induced subgraph obtained from $H$ by adding a universal vertex (when $t=4$ and $H$ is the three-vertex path, this yields their conjecture). The choice of $H$ in both result is best possible: (a) fails for every graph $H$ that is not $K_4$-free and chordal, and (b) fails for every graph $H$ that is not a forest.