Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let 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, Journal of Graph Theory, 24:297– 311, 1997] that, for every graph H, there is a function fH : N → R such that for every graph G ∈ Forb∗(H), χ(G) ≤ fH(ω(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.