Total proper connection of graphs

A graph is said to be total-colored if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph G, the total proper connection number of G, denoted by tpc(G), is defined as the smallest number of colors required to make G total-proper connected. These concepts are inspired by the concepts of proper connection number pc(G), proper vertex connection number pvc(G) and total rainbow connection number trc(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-connected graph G and give examples to show that the upper bound 4 is sharp. For general graphs, we also obtain an upper bound for tpc(G). Furthermore, we prove that tpc(G) ≤ 3n δ+1 + 1 for a connected graph G with order n and minimum degree δ. 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 tpc(G) and pc(G) can differ by t for 0 ≤ t ≤ 2.