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

TL;DR

This paper establishes upper bounds on the difference between the 2-hued chromatic number and the chromatic number of graphs, relating it to the independence number and other graph parameters.

Contribution

It provides new theoretical bounds for the 2-hued chromatic number in terms of independence number and graph degree parameters, extending previous understanding.

Findings

For regular graphs, the difference is at most logarithmic in the independence number.

For graphs with minimum degree at least 2, the difference involves a root and logarithmic terms.

In general, the difference is bounded by a function of the independence and clique numbers.

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 $\chi_2(G)$. 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 $.