On Acyclic Edge Colouring of Outerplanar Graphs etc
نویسندگان
چکیده
An acyclic edge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and is usually denoted by a′(G). Determining a ′(G) exactly is a very hard problem (both theoretically and algorithmically) and is not determined even for complete graphs. We show that a′(G) ≤ ∆(G) + 1, if G is an outerplanar graph. This bound is tight within an additive factor of 1 from optimality. Our proof is constructive leading to an O(n log∆) time algorithm. We also show that ∆ + 1 colours are sufficient for the class of fully subdevided graphs. Here, ∆ = ∆(G) denotes the maximum degree of the
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Acyclic Edge Colouring of Partial 2-Trees
An acyclic edge colouring of a graph is a proper edge colouring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and it is denoted by a′(G). Here, we obtain tight estimates on a′(G) for nontrivial subclasses of the family of 2-degenerate graphs. Specifically, we obtain values of...
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