Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].