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

Abstract

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 $v\in V(G)$ 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 {\it 2-hued chromatic number} of $G$ and denoted by $χ_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $ χ_{2}(G)- χ(G) \leq 2 \log _{2}(α(G)) +\mathcal{O}(1) $ and if $G$ is a graph and $δ(G)\geq 2$, then $ χ_{2}(G)- χ(G) \leq 1+\lceil \sqrt[δ-1]{4Δ^{2}} \rceil ( 1+ \log _{\frac{2Δ(G)}{2Δ(G)-δ(G)}} (α(G)) ) $ and in general case if $G$ is a graph, then $ χ_{2}(G)- χ(G) \leq 2+ \min \lbrace α^{\prime}(G),\frac{α(G)+ω(G)}{2}\rbrace $.