A note on Reed's conjecture for triangle-free graphs

Reed's Conjecture states that & chi;(G) & LE; ⠄(⠃(G) + & omega;(G) + 1)/2 ⠅, where & chi; (G), ⠃(G) and & omega;(G) are the chromatic number, maximum degree and clique number of a graph G, respectively. In this note, we prove this conjecture for maximal triangle-free graphs with maximum degree less than 7. Moreover, we show that Reed's Conjecture holds for all graphs with girth at least 5 up to at least 30 vertices and for all triangle-free graphs G up to at least 32 vertices such that & chi;(G) =⠆ 5 which improves similar results given in [Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, J. Graph Theory, 93(2020), 34-48.] & COPY; 2023 Elsevier B.V. All rights reserved.