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

A 2-hued coloring of a graph G is a coloring such that, for every vertex upsilon is an element of V (G) of degree at least 2, the neighbors of upsilon receive at least two 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 is denoted by chi(2) (G). In this paper, we will show that, if G is a regular graph, then chi(2) (C) - chi (G) <= 2log(2) (alpha(G)) + 3, and, if G is a graph and delta(G) >= 2, then chi(2) (G) - chi(G) <= 1 + inverted right perpendicular (delta-1)root 4 Delta(2) inverted left perpendicular (1 + log(2 Delta(G)/2 Delta(G) -) (delta(G)) (alpha(G))), and in the general case, if G is a graph, then chi(2)(G) - chi(G) <= 2 + min{alpha'(G), alpha(G)+omega(G)/2}. (C) 2012 Elsevier B.V. All rights reserved.