Acyclic edge colourings of graphs with the number of edges linearly bounded by the number of vertices |
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Authors: | Anna Fiedorowicz |
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Affiliation: | Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Z. Szafrana 4a, 65-516 Zielona Góra, Poland |
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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 .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 for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed. In general, the problem of computing the acyclic chromatic index of a graph is NP-complete. Only a few algorithms for finding acyclic edge colourings have been known by now. Moreover, these algorithms work only for graphs from particular classes.In our paper, we prove that for every graph G which satisfies the condition that |E(G′)|?t|V(G′)|−1 for each subgraph G′⊆G, where t?2 is a given integer, the constant p=2t3−3t+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. |
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Keywords: | Acyclic edge colouring Combinatorial problems Graph algorithms |
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