(Strong) Proper Connection in Some Digraphs

Abstract

An arc-colored digraph D is proper connected if any pair of vertices v i , v j ∈ V (D) there is a proper v i − v j path whose adjacent arcs have different colors and a proper v j − v i path whose adjacent arcs have different colors. The proper connection number of a digraph D is the minimum number of colors needed to make D proper connected, denoted by − → pc(D). An arc-colored digraph D is strong proper connected if any pair of vertices v i , v j ∈ V (D) there is a proper v i − v j geodesic and a proper v j − v i geodesic. The strong proper connection number of D is the minimum number of colors required to color the arcs of D in order to make D strong proper connected, denoted by − → spc(D). In this paper, we will show some results on − → pc(D) and − → spc(D), mostly for the case of the (strong) proper connection numbers of cacti and circulant digraphs. INDEX TERMS Proper path, proper connection number, proper geodesic, strong proper connection number.

Proper connection and proper-walk connection of digraphs

“…. , n − 1}, |S| ≥ 2 and 1 ∈ S. This result implies a theorem proved in [14], which states that − → pc(C n ([k]) = 2 whenever k = n − 1 and k = 1. In Section 5, we give some sufficient conditions for a Hamiltonian digraph to have proper-walk connection number equal to 2.…”

“…The directed version of the proper connection was introduced by Magnant et al in [15], and the directed version of the proper-walk connection by Melville and Goddard in [16]. In [15] and in [14] the strong and the vertex version of directed proper connection were considered. In this paper we study the proper connection and the proper-walk connection of digraphs.…”