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

Abstract

Reed's Conjecture states that χ(G)≤⌈(Δ(G)+ω(G)+1)/2⌉, where χ(G), Δ(G) and ω(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 χ(G)≠5 which improves similar results given in [Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, J. Graph Theory, 93(2020), 34–48.]