Path coloring on binary caterpillars

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 [8/5L] colors, where L is the maximum number of paths sharing an 5 edge. Furthermore, we show that no local greedy path coloring algorithm on caterpillars in general uses less than [8/5L] colors.