Proper Vertex Connection and Total Proper Connection
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Notions of vertex proper connection, the vertex-coloring version of the proper connection number, have been defined and studied independently in Chizmar et al. (AKCE Int J Graphs Comb 13(2):103–106, 2016) and Jiang et al. (Bull Malays Math Sci Soc 41(1):415–425, 2018). A vertex-colored graph G is called proper vertex k-connected if every pair of vertices is connected by k internally disjoint paths, each of which has no two consecutive internal vertices of the same color. Define the proper vertex k-connection number of G, denoted by pvc k (G), to be the smallest number of (vertex) colors needed to make G proper vertex k-connected. We write pvc(G) for pvc1(G). Here the end vertices are not included to be consistent with the similarly defined rainbow vertex connection number where, if end vertices were included, all vertices would necessarily receive distinct colors.