Chromatic numbers of exact distance graphs

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

Research output: Contribution to journalArticle

3 Scopus citations

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

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

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