Excluding induced subdivisions of the bull and related graphs

For any graph $H$, let ${\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{b}}^{\ast }(H)$ 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}$ \rightarrow $\mathbb{R}$$suchthatforeverygraph$G \in {\rm Forb}^*(H)$,$\chi(G) \leq f_H(\omega(G))$\mathrm{.}Weprovethisconjectureforseveralgraphs$H$, 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.