Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T19:15:50.811Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

[1] de Grey, A. D. N. J. (2018) The chromatic number of the plane is at least 5. arXiv:1804.02385Google Scholar
[2] van den Heuvel, J., Kierstead, H. A. and Quiroz, D. (2016) Chromatic numbers of exact distance graphs. J. Combin. Theory Ser. B. arXiv:1612.02160Google Scholar
[3] Kloeckner, B. R. (2015) Coloring distance graphs: A few answers and many questions. Geombinatorics 24 117–134.Google Scholar
[4] Nešetřil, J. and Ossona de Mendez, P. (2012) Sparsity: Graphs, Structures, and Algorithms, Springer.Google Scholar
[5] Nešetřil, J. and Ossona de Mendez, P. (2015) On low tree-depth decompositions. Graphs Combin. 31 1941–1963.Google Scholar
[6] Parlier, H. and Petit, C. (2017) Chromatic numbers for the hyperbolic plane and discrete analogs. arXiv:1701.08648Google Scholar
[7] Parlier, H. and Petit, C. (2016) Chromatic numbers of hyperbolic surfaces. Indiana Univ. Math. J. 65 1401–1423.Google Scholar
[8] Quiroz, D. (2017) Colouring exact distance graphs of chordal graphs. arXiv:1703.07008Google Scholar