Excluding induced subdivisions of the bull and related graphs

TL;DR

This paper proves a conjecture relating to the chromatic number of graphs avoiding induced subdivisions of specific graphs, including the paw, bull, and a new 'necklace' graph, establishing bounds based on clique number.

Contribution

It confirms the conjecture for several graphs by showing that graphs avoiding induced subdivisions of these graphs have bounded chromatic number in terms of their clique number.

Findings

Confirmed the conjecture for the paw, bull, and necklace graphs.

Established that graphs with no induced subdivision of these graphs have bounded chromatic number.

Provided new classes of graphs for which the chromatic bound depends on clique number.

Abstract

For any graph $H$, let ${\rm Forb}^*(H)$ 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 $f_H:\mathbb{N} \rightarrow \mathbb{R}$ such that for every graph $G \in {\rm Forb}^*(H)$, $\chi(G) \leq f_H(\omega(G))$. 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.