Polynomial $\chi$-binding functions for $t$-broom-free graphs

For any positive integer t, a t-broom is a graph obtained from K1,t+1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t-brooms, we have χ(G) = o(ω(G)), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. When t = 2, this answers a question of Schiermeyer and Randerath. Moreover, for t = 2, we strengthen the bound on χ(G) to 7.5ω(G), confirming a conjecture of Sivaraman. For t ≥ 3 and {t-broom,Kt,t}-free graphs, we improve the bound to o(ω t−1+ 2 t+1 ).