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

TL;DR

The paper proves that every even-hole-free graph of sufficiently large treewidth contains a four-vertex induced subgraph with at least five edges, namely a $K_4$ or a diamond, resolving a conjecture of Sintiari and Trotignon. It strengthens this by showing that for any $K_4$-free chordal $H$, such graphs contain either a $K_4$ or an induced subgraph isomorphic to $H$, and for any forest $H$ and any $t$, they contain a $t$-clique or a cone over $H$. The approach reduces to embedding $2$-trees inside the host graphs, using contraptions to preserve the class $oxed{ ext{E}}$ under minors, and a kaleidoscope/mirroring framework to iteratively grow blurry copies into actual $2$-trees. Consequently, for fixed $t$, $( ext{even hole}, ext{diamond}, K_t)$-free graphs have bounded treewidth, and cones over trees (i.e., $ ext{cone}(T)$) arise as the induced subgraphs guaranteed by large treewidth in the broader class, providing a detailed local-structure picture of these graphs.

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.

Figures (12)

• Figure 1: The $4$-basic obstructions, with a subdivided $4$-by-$4$ wall in the middle and its line graph on the right.
• Figure 2: The $3$-basic obstructions other than the complete graph $K_4$, featuring a theta in $K_{3,3}$ (left), a theta in a subdivision of $W_{3\times 3}$ (middle), and a prism in the line graph of the same subdivided $W_{3\times 3}$ (right). An even hole in each theta and prism is highlighted. (See \ref{['sec:defns']} for the definitions.)
• Figure 3: A coned tree as in Theorems \ref{['thm:mainforest+']} (left) and \ref{['thm:tree+']} (middle), and an arbitrary $2$-tree (right).
• Figure 4: Proof of Theorem \ref{['thm:k-tree']} (when $k=5$ and $|N|=2$).
• Figure 5: From left to right, a theta, a prism and an even wheel. Squiggly lines represent paths of arbitrary length (possibly zero).

Theorems & Definitions (48)

• Theorem 1.1: Robertson and Seymour GMV
• Theorem 1.2: Sintiari and Trotignon layered-wheels
• Conjecture 1.3: Sintiari and Trotignon layered-wheels
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• proof
• Theorem 1.9
• Theorem 1.10