Polynomial ?-binding functions for t-broom-free graphs

For any positive integer t, a t-broomis a graph obtained from K1,t+1by subdividing an edge once. In this paper, we show that, for graphs Gwithout induced t-brooms, we have.(G) = o(.(G)t+1), where.(G) and.(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.(G) to 7.(G)2, confirming a conjecture of Sivaraman. For t = 3and {t-broom, Kt,t}-free graphs, we improve the bound to o(.t). (c) 2023 Elsevier Inc. All rights reserved.