Acyclic edge colourings of graphs with the number of edges linearly bounded by the number of vertices

  • Authors:

  • Anna Fiedorowicz

  • Affiliations:

  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Z. Szafrana 4a, 65-516 Zielona Góra, Poland

  • Venue:
  • Information Processing Letters
  • Year:
  • 2011

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Abstract

Let G be any finite graph. A mapping c:E(G)-{1,...,k} is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges that have colour i or j is acyclic. The smallest number k of colours such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by @g"a^'(G). Determining the acyclic chromatic index of a graph is a hard problem, both from theoretical and algorithmical point of view. In 1991, Alon et al. proved that @g"a^'(G)==2 is a given integer, the constant p=2t^3-3t+2. Based on that result, we obtain a polynomial algorithm which computes such a colouring. The class of graphs covered by our theorem is quite rich, for example, it contains all t-degenerate graphs.