Path Coloring on Binary Caterpillars


IEICE TRANSACTIONS on Information and Systems   Vol.E89-D   No.6   pp.1906-1913
Publication Date: 2006/06/01
Online ISSN: 1745-1361
DOI: 10.1093/ietisy/e89-d.6.1906
Print ISSN: 0916-8532
Type of Manuscript: PAPER
Category: Algorithm Theory
path coloring,  wavelength routing,  caterpillar,  

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The path coloring problem is to assign the minimum number of colors to a given set P of directed paths on a given symmetric digraph D so that no two paths sharing an arc have the same color. The problem has applications to efficient assignment of wavelengths to communications on WDM optical networks. In this paper, we show that the path coloring problem is NP-hard even if the underlying graph of D is restricted to a binary caterpillar. Moreover, we give a polynomial time algorithm which constructs, given a binary caterpillar G and a set P of directed paths on the symmetric digraph associated with G, a path coloring of P with at most colors, where L is the maximum number of paths sharing an edge. Furthermore, we show that no local greedy path coloring algorithm on caterpillars in general uses less than colors.