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The Distance-t Chromatic Index of Graphs

Published online by Cambridge University Press:  14 November 2013

TOMÁŠ KAISER  and
ROSS J. KANG
TOMÁŠ KAISER
Affiliation:
Department of Mathematics, Institute for Theoretical Computer Science (CE–ITI), and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Plzeň, Czech Republic (e-mail: [email protected])
ROSS J. KANG
Affiliation:
Mathematical Institute, Utrecht University, Utrecht, Netherlands (e-mail: [email protected])
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Abstract

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

Keywords

Primary 05C15Secondary 05C3505D4005C70
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 90 - 101
Copyright
Copyright © Cambridge University Press 2013 

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