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 [email protected]?V(G) of degree at least 2, the neighbors of v 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 @g"2(G). In this paper, we will show that, if G is a regular graph, then @g"2(G)[email protected](G)@?2log"2(@a(G))+3, and, if G is a graph and @d(G)>=2, then @g"2(G)[email protected](G)@[email protected][email protected]^[email protected]@?(1+log"2"@D"("G")"2"@D"("G")"-"@d"("G")(@a(G))), and in the general case, if G is a graph, then @g"2(G)[email protected](G)@?2+min{@a^'(G),@a(G)[email protected](G)2}.

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