On the strong chromatic number of graphs

TL;DR

This paper investigates the strong chromatic number of graphs, providing improved bounds for graphs with large maximum degree and demonstrating the sharpness of these bounds.

Contribution

The authors improve the upper bound of the strong chromatic number for graphs with large maximum degree, showing it is at most 2Δ when Δ ≥ n/6, and prove this bound is tight.

Findings

Bound of 2Δ for large degree graphs is sharp

Improved upper bound from 3Δ-1 to 2Δ for certain graphs

Established tightness of the new bound

Abstract

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph into disjoint subsets $V_1, \ldots, V_{\lceil n/k\rceil}$ of size $k$ each, one can find a proper $k$-vertex-coloring of the graph such that each part $V_i$, $i=1, \ldots, \lceil n/k\rceil$, contains exactly one vertex of each color. For any graph $G$ with maximum degree $\Delta$, it is easy to see that $\chi_S(G)\geq\Delta+1$. Recently, Haxell proved that $\chi_S(G) \leq 3\Delta -1$. In this paper, we improve this bound for graphs with large maximum degree. We show that $\chi_S(G)\leq 2\Delta$ if $\Delta \geq n/6$ and prove that this bound is sharp.