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An asymptotic bound for the strong chromatic number

Part of: Graph theory

Published online by Cambridge University Press:  15 March 2019

Allan Lo  and
Nicolás Sanhueza-Matamala
Allan Lo
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
Nicolás Sanhueza-Matamala*
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
*
*Corresponding author. Email: [email protected]
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Abstract

The strong chromatic number χs(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 Kr 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) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 768 - 776
Copyright
© Cambridge University Press 2019 

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Footnotes

†

The research leading to these results was partially supported by EPSRC, grant no. EP/P002420/1.

‡

The research leading to these results was partially supported by the Becas Chile scholarship scheme from CONICYT.

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