Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 463–472 | Cite as

Properly colored trails, paths, and bridges



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.


Edge coloring Properly colored trails Path connection number Bridges 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA

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