On an upper bound of a graph's chromatic number, depending on the graph's degree and density

AbstractGrünbaum's conjecture on the existence of k-chromatic graphs of degree k and girth g for every k ≥ 3, g ≥ 3 is disproved. In particular, the bound obtained states that the chromatic number of a triangle-free graph does not exceed [3(σ + 2)4], where σ is the graph's degree