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 $pvc_{k}(G)$, 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 $rvc_k(G)$, strong rainbow vertex-connection number $srvc(G)$, and proper $k$-connection number $pc_k(G)$ of a $k$-connected graph $G$. Firstly, we determine the value of $pvc(G)$ for general graphs and $pvc_k(G)$ for some specific graphs. We also compare the values of $pvc_k(G)$ and $pc_k(G)$. Then, sharp bounds of $spvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq spvc(G)\leq n-2$. 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 $\chi(G)$ of a connected graph $G$.