On the Total Proper Connection of Graphs

The total-coloring of a graph is a coloring of the edge set and vertex set. A path in a total-colored graph is a total proper path if the coloring of the edges and internal vertices is proper, that is, any two adjacent or incident elements of edges and internal vertices on the path differ in color. For a connected graph G, the total proper connection number of G, denoted by tpc(G), is defined as the smallest number of colors such that any two vertices of the graph are connected by a total proper path of G. These concepts are inspired by the concepts of total chromatic number chi ''(G), proper connection number pc(G) and proper vertex connection number pvc(G) of a connected graph G. In this paper, we first determine the value of the total proper connection number tpc(G) for some special graphs G. Secondly, we obtain that tpc(G)<= 4 for any 2-(edge-)connected graph G and give examples to show that the upper bound 4 is sharp. For general graphs, we also obtain sharp upper bounds for tpc(G) in terms of the maximum degree of a vertex incident with a bridge in G and the minimum degree, respectively. Finally, we compare tpc(G) with pvc(G) and pc(G), respectively, and obtain that tpc(G)>pvc(G) for any nontrivial connected graph G and that there exist graphs G such that tpc(G)=pc(G)+t for 0 <= t <= 2.