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.
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