On (Strong) Proper Vertex-Connection of Graphs

Abstract

A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is 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 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 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 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, $0\le spvc(G)\le n-2$. 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.