An asymptotic bound for the strong chromatic number

TL;DR

This paper establishes an asymptotic upper bound for the strong chromatic number of graphs with large maximum degree, showing it is at most approximately twice the maximum degree, which is nearly optimal.

Contribution

It proves a tight asymptotic bound for the strong chromatic number in graphs with linear maximum degree, advancing understanding of graph coloring constraints.

Findings

Strong chromatic number is at most approximately twice the maximum degree for large graphs.

The bound is asymptotically optimal, matching known lower bounds.

The result applies to graphs with minimum degree proportional to the number of vertices.

Abstract

The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of spanning disjoint copies of $K_r$ in the same vertex set, the resulting graph has a proper vertex-colouring with $r$ colours. We show that for every $c > 0$ and every graph $G$ on $n$ vertices with $\Delta(G) \ge cn$, $\chi_{\text{s}}(G) \leq (2 + o(1)) \Delta(G)$, which is asymptotically best possible.