TY - JOUR

T1 - Chromatic numbers of exact distance graphs

AU - van den Heuvel, Jan

AU - Kierstead, Henry

AU - Quiroz, Daniel A.

N1 - Funding Information:
The research for this paper was started during a stay of the first two authors at the Mittag-Leffler Institute in Stockholm. JvdH and HAK would like to thank the Mittag-Leffler Institute for hospitality and support. DAQ thankfully acknowledges support from CONICYT , PIA/Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal AFB170001 .
Funding Information:
The research for this paper was started during a stay of the first two authors at the Mittag-Leffler Institute in Stockholm. JvdH and HAK would like to thank the Mittag-Leffler Institute for hospitality and support. DAQ thankfully acknowledges support from CONICYT, PIA/Concurso Apoyo a Centros Cient?ficos y Tecnol?gicos de Excelencia con Financiamiento Basal AFB170001.
Funding Information:
The research for this paper was started during a stay of the first two authors at the Mittag-Leffler Institute in Stockholm. JvdH and HAK would like to thank the Mittag-Leffler Institute for hospitality and support. DAQ thankfully acknowledges support from CONICYT, PIA/Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal AFB170001.
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2019/1

Y1 - 2019/1

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

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U2 - 10.1016/j.jctb.2018.05.007

DO - 10.1016/j.jctb.2018.05.007

M3 - Article

AN - SCOPUS:85047953697

SN - 0095-8956

VL - 134

SP - 143

EP - 163

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -