A NOTE ON THE PROPER CONFLICT-FREE CONNECTION NUMBER OF CONNECTED GRAPHS

Abstract

Let ???? be an edge-colored graph. A path in ???? is a conflict-free path if it contains a color used on exactly one of its edges. The graph ???? is called conflict-free connected if every two distinct vertices is connected by at least one conflict-free path. The graph ???? is said to be properly edge colored (properly colored for simplifying), meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. If graph ???? is simultaneously properly colored and conflict-free connected, then Czap et al. [1] introduced the concept of properly conflict-free connected. The proper conflict-free connection number, denoted by ????????????????(????), is the minimum number of colors needed in order to make it properly conflict-free connected. Recently, there is few results of proper conflict[1]free connection. In this paper, we determine some connected graph classes having ????????????????(????) = ????′(????).