On (strong) proper vertex-connection of graphs
A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex k-connected} if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the {\it proper vertex k-connection number} of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is {\it strong proper vertex-connected}, if for any two vertices u,v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the {\it strong proper vertex-connection number} of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. These concepts are inspired by the concepts of rainbow vertex k-connection number rvck(G), strong rainbow vertex-connection number srvc(G), and proper k-connection number pck(G) of a k-connected graph G. Firstly, we determine the value of pvc(G) for general graphs and pvck(G) for some specific graphs. We also compare the values of pvck(G) and pck(G). Then, sharp bounds of spvc(G) are given for a connected graph G of order n, that is, 0≤spvc(G)≤n−2. Moreover, we characterize the graphs of order n such that spvc(G)=n−2,n−3, respectively. Finally, we study the relationship among the three vertex-coloring parameters, namely, spvc(G), srvc(G) and the chromatic number χ(G) of a connected graph G.