The permutation-path coloring problem on trees

In this paper we rst show that the permutation-path coloring problem is NP-hard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations which are permutations that contain exactly one cycle on trees with maximum degree greater or equal to 4. We calculate a lower bound on the average complexity of the permutation-path coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutation-path coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n=4 + o(n). We extend these results and obtain an upper bound on the average complexity of the permutation-path coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutions-path coloring problem on arbitrary trees.