An asymptotic bound for the strong chromatic number

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 $r \lceil n/r \rceil - n$isolatedverticesto$G$ 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 $\mathrm{\Delta }(G)\ge cn$, ${\chi }_{\text{s}}(G)\le (2+o(1))\mathrm{\Delta }(G)$, which is asymptotically best possible.