Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Abstract: Gy\'arf\'as and Sumner independently conjectured that for every tree $T$, the class of graphs not containing $T$ as an induced subgraph is $\chi$-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a function of their clique numbers. This remains open for general trees $T$, but has been proved for some particular trees. For $k\ge 1$, let us say a broom of length $k$ is a tree obtained from a $k$-edge path with ends $a,b$ by adding some number of leaves adjacent to $b$, and we call $a$ its handle. A tree obtained from brooms of lengths $k_1,...,k_n$ by identifying their handles is a $(k_1,...,k_n)$-multibroom. Kierstead and Penrice proved that every $(1,...,1)$-multibroom $T$ satisfies the Gy\'arf\'as-Sumner conjecture, and Kierstead and Zhu proved the same for $(2,...,2)$-multibrooms. In this paper give a common generalization: we prove that every $(1,...,1,2,...,2)$-multibroom satisfies the Gy\'arf\'as-Sumner conjecture.