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Exact Distance Colouring in Trees

Part of: Graph theory

Published online by Cambridge University Press:  24 July 2018

NICOLAS BOUSQUET ,
LOUIS ESPERET ,
ARARAT HARUTYUNYAN  and
RÉMI DE JOANNIS DE VERCLOS
NICOLAS BOUSQUET
Affiliation:
Université Grenoble Alpes, CNRS, G-SCOP, Grenoble, France (e-mail: nicolas.bousquet@grenoble-inp.fr, louis.esperet@grenoble-inp.fr)
LOUIS ESPERET
Affiliation:
Université Grenoble Alpes, CNRS, G-SCOP, Grenoble, France (e-mail: nicolas.bousquet@grenoble-inp.fr, louis.esperet@grenoble-inp.fr)
ARARAT HARUTYUNYAN
Affiliation:
LAMSADE, University of Paris-Dauphine, Paris, France (e-mail: ararat.harutyunyan@dauphine.fr)
RÉMI DE JOANNIS DE VERCLOS
Affiliation:
Radboud University Nijmegen, Netherlands (e-mail: r.deverclos@math.ru.nl)
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Abstract

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 177 - 186
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

The authors were partially supported by ANR Projects STINT (anr-13-bs02-0007) and GATO (anr-16-ce40-0009-01), and LabEx PERSYVAL-Lab (anr-11-labx-0025-01) and LabEx CIMI

References

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