Properly colored trails, paths, and bridges

Abstract

The proper-trail connection number of a graph is the minimum number of colors needed to color the edges such that every pair of vertices are joined by a trail without two consecutive edges of the same color; the proper-path connection number is defined similarly. In this paper we consider these in both bridgeless graphs and graphs in general. The main result is that both parameters are tied to the maximum number of bridges incident with a vertex. In particular, we provide for $k\ge 4$ a simple characterization of graphs with proper-trail connection number k, and show that the proper-path connection number can be approximated in polynomial-time within an additive 2.