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 $\chi_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $ \chi_{ 2}(G)- \chi(G) \leq 2 \log _{ 2}(\alpha(G)) +\mathcal{ O}(1) $ and if $G$ is a graph and $\delta(G)\geq 2$, then $ \chi_{ 2}(G)- \chi(G) \leq 1+\lceil \sqrt[\delta -1]{ 4\Delta^{ 2}} \rceil ( 1+ \log _{ \frac{ 2\Delta(G)}{ 2\Delta(G)-\delta(G)}} (\alpha(G)) ) $ and in general case if $G$ is a graph, then $ \chi_{ 2}(G)- \chi(G) \leq 2+ \min \lbrace \alpha^{ \prime}(G),\frac{ \alpha(G)+\omega(G)}{ 2}\rbrace $.
Categories: math.CO cs.DM