Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

A 2-hued coloring of a graph GG is a coloring such that, for every vertex v∈V(G)v∈V(G) of degree at least 22, the neighbors of vv receive at least two colors. The smallest integer kk such that GG has a 2-hued coloring with kk colors is called the 2-hued chromatic number   of GG, and is denoted by χ2(G)χ2(G). In this paper, we will show that, if GG is a regular graph, then χ2(G)−χ(G)≤2log2(α(G))+3χ2(G)−χ(G)≤2log2(α(G))+3, and, if GG is a graph and δ(G)≥2δ(G)≥2, then χ2(G)−χ(G)≤1+⌈4Δ2δ−1⌉(1+log2Δ(G)2Δ(G)−δ(G)(α(G))), and in the general case, if GG is a graph, then χ2(G)−χ(G)≤2+min{α′(G),α(G)+ω(G)2}.