Chromatic numbers of exact distance graphs

Jan van den Heuvel, Henry Kierstead, Daniel A. Quiroz

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.

Original languageEnglish (US)
JournalJournal of Combinatorial Theory. Series B
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Distance Graph
Coloring
Chromatic number
Colouring
Graph in graph theory
Odd
Color
Lower bound
Integer
Graph Classes
Maximum Degree
Minor
If and only if
Vertex of a graph

Keywords

  • Bounded expansion
  • Chromatic number
  • Exact distance graphs
  • Generalised colouring numbers
  • Planar graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

Chromatic numbers of exact distance graphs. / van den Heuvel, Jan; Kierstead, Henry; Quiroz, Daniel A.

In: Journal of Combinatorial Theory. Series B, 01.01.2018.

Research output: Contribution to journalArticle

@article{f0a271c5e5e949188f8b37e7bb647575,
title = "Chromatic numbers of exact distance graphs",
abstract = "For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.",
keywords = "Bounded expansion, Chromatic number, Exact distance graphs, Generalised colouring numbers, Planar graphs",
author = "{van den Heuvel}, Jan and Henry Kierstead and Quiroz, {Daniel A.}",
year = "2018",
month = "1",
day = "1",
doi = "10.1016/j.jctb.2018.05.007",
language = "English (US)",
journal = "Journal of Combinatorial Theory. Series B",
issn = "0095-8956",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Chromatic numbers of exact distance graphs

AU - van den Heuvel, Jan

AU - Kierstead, Henry

AU - Quiroz, Daniel A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.

AB - For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.

KW - Bounded expansion

KW - Chromatic number

KW - Exact distance graphs

KW - Generalised colouring numbers

KW - Planar graphs

UR - http://www.scopus.com/inward/record.url?scp=85047953697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047953697&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2018.05.007

DO - 10.1016/j.jctb.2018.05.007

M3 - Article

AN - SCOPUS:85047953697

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -