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

A 2-hued coloring of a graph G (also known as conditional (k, 2)-coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a 2-hued coloring with k colors, is called the 2-hued chromatic number of G and denoted by χ2(G). In this paper, we will show that if G is a regular graph, then χ2(G)− χ(G) ≤ 2 log2(α(G)) +O(1) and if G is a graph and δ(G) ≥ 2, then χ2(G)− χ(G) ≤ 1 + ⌈ δ−1 √ 4∆2⌉(1 + log 2∆(G) 2∆(G)−δ(G) (α(G))) and in general case if G is a graph, then χ2(G)− χ(G) ≤ 2 + min{α′(G), α(G)+ω(G) 2 }.