Excluding induced subdivisions of the bull and related graphs

Abstract

For any graph $H$, let ${\mathrm{Forb}}^{\ast }(H)$ be the class of graphs with no induced subdivision of $H$. It was conjectured in [A.D. Scott, Induced trees in graphs of large chromatic number, {\em Journal of Graph Theory}, 24:297--311, 1997] that, for every graph $H$, there is a function $f_H:\mathbb{N}\to \mathbb{R}$ such that for every graph $G\in {\mathrm{Forb}}^{\ast }(H)$, $\chi (G)\le f_H(\omega (G))$. We prove this conjecture for several graphs $H$, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex-disjoint pendant edges), and what we call a "necklace," that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge.

References in corpus (1)

• A structure theorem for graphs with no cycle with a unique chord and its consequences

Cited by in corpus (2)

• Triangle-free graphs that do not contain an induced subdivision of $K_4$ are 3-colorable
• Structure of (bull, diamond)-free graphs and its applications