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

For any positive integer $t$, a emph{$t$-broom} is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $chi(G) = o(omega(G)^{t+1})$, where $chi(G)$ and $omega(G)$ 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 $chi(G)$ to $7.5omega(G)^2$, confirming a conjecture of Sivaraman. For $tgeq 3$ and {$t$-broom, $K_{t,t}$}-free graphs, we improve the bound to $o(omega^{t-1+frac{2}{t+1}})$.