Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth

TL;DR

This paper advances the understanding of when even-hole-free graphs of large treewidth must contain a fixed structure by formulating and proving two main results: a common strengthening that handles cones on forests and 2-forests (Theorem maincone1) and a result proving the conjecture for crystals (Theorem maincrystal1). The authors introduce phantom layered wheels and develop a framework tying large-treewidth graphs to induced subgraphs via Ramsey-type arguments, enabling the construction of crystals or cone gadgets from phantoms. They situate these results within a broader sparse-obstruction viewpoint, discuss related conjectures, and note that while some stronger conjectures have been refuted, the central conjecture remains open. The work blends structural graph theory with detailed gadgetry and Ramsey theory to connect global treewidth with local induced subgraph structure, offering a pathway toward a grid-like characterization in even-hole-free graphs.

Abstract

We present and study the following conjecture: for an integer $t\geq 4$ and a graph $H$, every even-hole-free graph of large enough treewidth has an induced subgraph isomorphic to either $K_t$ or $H$, if (and only if) $H$ is a $K_4$-free chordal graph. The ``only if'' part follows from the properties of the so-called layered wheels due to Sintiari and Trotignon. Alecu, Chudnovsky, Spirkl and the author recently proved the conjecture in two special cases: (a) when $t=4$; and (b) when $H=cone (F)$ for some forest $F$; that is, $H$ is obtained from $F$ by adding a universal vertex. Our first result is a common strengthening: for an integer $t\geq 4$ and graphs $F$ and $H$, (even-hole, $cone(cone (F))$, $H$, $K_t$)-free graphs have bounded treewidth if and only if $F$ is a forest and $H$ is a $K_4$-free chordal graph. Also, for general $t\geq 4$, we push the current state of the art further than (b) by settling the conjecture for the smallest choices of $H$ that are not coned forests. This follows from our second result: we prove the conjecture when $H$ is a crystal; that is, a graph obtained from several coned double stars by gluing them together along the middle edges of the double stars. In the first version of this paper, we suggested a strengthening of our main conjecture, that for every $t\geq 1$, every graph of sufficiently large treewidth has an induced subgraph of treewidth $t$ which is either complete, complete bipartite, or $2$-degenerate. This strengthening has now been refuted by Chudnovsky and Trotignon [On treewidth and maximum cliques, arXiv:2405.07471, 2024].

Figures (13)

• Figure 1: The graph $W_{5\times 5}$.
• Figure 2: The $4$-basic obstructions (the graph on the right is the line graph of the subdivided wall in the middle).
• Figure 3: For $t\geq 3$, all non-complete $t$-basic obstructions have even holes. Note, in thick lines, the theta on the left and the middle, and the prism on the right, and note the even holes highlighted in them.
• Figure 4: From left to right: the diamond, a coned tree, the gem, and the two smallest ($2$-connected) $2$-forests that are not coned forests (the last one but one is commonly known as the antinet).
• Figure 5: A crystal.

Theorems & Definitions (49)

• Conjecture 1.1
• Theorem 1.2: Robertson and Seymour GMV
• Theorem 1.5: Sintiari and Trotignon layered-wheels
• Conjecture 1.7
• Conjecture 1.8
• Theorem 1.9: Alecu, Chudnovsky, Hajebi and Spirkl twxi
• Theorem 1.10: Alecu, Chudnovsky, Hajebi and Spirkl twxi
• Theorem 1.11
• Theorem 1.12
• Conjecture 1.13