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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 patharXiv:1512.00726v1 fatcat:3qalrlhfwfgntl42skgtaz2kcy