A positive instance of Scott's Conjecture on induced subdivisions

TL;DR

The paper proves a new χ-boundedness result for Scott's conjecture on induced subdivisions by focusing on graphs $H$ that are a complete bipartite graph together with an extra vertex adjacent to exactly two vertices in one part, i.e., the family $\mathcal{P}_{a,a}$ with $a\ge 2$. The authors develop a template method that expands a large complete bipartite core into a complete multipartite core and then bound the chromatic number by analyzing how the remaining vertices attach, using Narayanan-type connectivity results and structural lemmas. The main contribution is showing that IS$\mathcal{P}_{a,a}$-free graphs are χ-bounded, with concrete corollaries including IS$K_4^{++}$-free graphs, thereby advancing the understanding of Scott's conjecture for new H-forms. While the current bound is not polynomial in the parameters due to the employed techniques, the result identifies a broad new family of graphs for which induced-subdivision χ-boundedness holds, enriching the landscape of χ-bounded hereditary classes.

Abstract

For a graph $G$, $χ(G)$ denotes the chromatic number of $G$ and $ω(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $χ$-bounded if there is a function $f$ such that for each graph $G$ in the class, $χ(G) \le f(ω(G))$. Scott (1997) conjectured that for every graph $H$, the class of graphs which do not contain any subdivision of $H$ as an induced subgraph is $χ$-bounded. He proved his conjecture when $H$ is a tree and when $H$ is the complete graph on four vertices, $K_4$. Esperet and Trotignon (2019) proved that the conjecture holds when $H$ is $K_4$ with one edge subdivided once. Scott's conjecture was disproved by Pawlik et al. (2014). Chalopin et al. (2016) gave more counterexamples including the graph obtained from $K_4$ by subdividing each edge of a 4-cycle once. We prove that the conjecture holds when $H$ consists of a complete bipartite graph with and additional vertex which has exactly two neighbours, on the same side of the bipartition. As a special case, this proves Scott's conjecture when $H$ is obtained from $K_4$ by subdividing two disjoint edges.

Theorems & Definitions (19)

• Theorem 1: Kühn and Osthus, KuhnOsthus
• Theorem 2: Bourneuf, Bucić, Cook, and Davies polyko1, Girão and Hunter polyko2
• Theorem 3: Nguyen Nguyen
• Theorem 4: Alon, Pach, Solymosi aps
• Theorem 5
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3