# A linear algorithm for edge-coloring partial *k*-trees

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## Abstract

Many combinatorial problems can be efficiently solved for partial *k*-trees. The edge-coloring problem is one of a few combinatorial problems for which no linear-time algorithm has been obtained for partial *k*-trees. The best known algorithm solves the problem for partial *k*-trees *G* in time \(O\left( {n\Delta ^{2^{2\left( {k + 1} \right)} } } \right)\) where *n* is the number of vertices and Δ is the maximum degree of *G*. This paper gives a linear algorithm which optimally edge-colors a given partial *k*-tree for fixed *k*.

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© Springer-Verlag Berlin Heidelberg 1993