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arXiv · 1711.08214

An asymptotic bound for the strong chromatic number

Allan Lo·Nicolás Sanhueza-Matamala
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Abstract

The strong chromatic number $χ_{\text{s}}(G)$ 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 $Δ(G) \ge cn$, $χ_{\text{s}}(G) \leq (2 + o(1)) Δ(G)$, which is asymptotically best possible.

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Allan Lo, Nicolás Sanhueza-Matamala. 2018-08-09. An asymptotic bound for the strong chromatic number. https://doi.org/10.1017/s0963548318000561

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