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arXiv · 2602.11990

A positive instance of Scott's Conjecture on induced subdivisions

Kathie Cameron·Ni Luh Dewi Sintiari·Sophie Spirkl
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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.

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Kathie Cameron, Ni Luh Dewi Sintiari, Sophie Spirkl. 2026-02-12. A positive instance of Scott's Conjecture on induced subdivisions. https://arxiv.org/abs/2602.11990

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