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- W1988145916 abstract "Let G=(V,E) be any finite graph. A mapping C:E→[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 which 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, denoted by χa′(G). In 2001, Alon et al. conjectured that for any graph G it holds that χa′(G)≤Δ(G)+2; here Δ(G) stands for the maximum degree of G. In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that χa′(G)≤Δ(G)+1 if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then χa′(G)≤Δ(G)+15." @default.
- W1988145916 created "2016-06-24" @default.
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- W1988145916 date "2010-05-01" @default.
- W1988145916 modified "2023-10-05" @default.
- W1988145916 title "Acyclic edge colouring of planar graphs without short cycles" @default.
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- W1988145916 doi "https://doi.org/10.1016/j.disc.2009.06.007" @default.
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