A positive instance of Scott's Conjecture on induced subdivisions

TL;DR

This paper proves Scott's Conjecture for a specific class of graphs, showing that graphs excluding certain induced subdivisions are χ-bounded, thus advancing understanding of graph coloring related to induced subgraph restrictions.

Contribution

The paper establishes Scott's Conjecture for graphs where H is a bipartite graph with an extra vertex connected to two vertices on the same side, including a case related to subdivided K4.

Findings

Scott's Conjecture holds for the specified class of graphs.

Identifies a new family of graphs satisfying χ-boundedness.

Extends the class of graphs for which the conjecture is verified.

Abstract

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in the class, $\chi(G) \le f(\omega(G))$. 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 $\chi$-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.