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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.11990 |
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| _version_ | 1866910020500717568 |
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| author | Cameron, Kathie Sintiari, Ni Luh Dewi Spirkl, Sophie |
| author_facet | Cameron, Kathie Sintiari, Ni Luh Dewi Spirkl, Sophie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11990 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A positive instance of Scott's Conjecture on induced subdivisions Cameron, Kathie Sintiari, Ni Luh Dewi Spirkl, Sophie Combinatorics 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. |
| title | A positive instance of Scott's Conjecture on induced subdivisions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.11990 |