Several Special Proper Total Colorings Of Graphs

In this paper, we discuss several special proper total colorings of graphs, including adjacent vertex-distinguishing total coloring, D(β)-vertex-distinguishing total coloring andβ-frugal total coloring.In section one, the exact numbers of 2-connected outerplanar graphs with maximum degree 5 are given; In section two, circular graphs C_n(1,k) are exactly vertex-distinguishingly totally colored when k equals to three, four, (n-1)/2(n is odd), respectively.In section three, a D(2)-vertex-distinguishing total chromatic number of trees is obtained by virtue of Hall's Theorem in combinatorial mathematics. In section four, a new definition,αβ-frugal total coloring of graphs, is proposed and the general form of Lavasz Lodal Lemma in probabilistic theory is used to get one upper bound of theβ-frugal total chromatic number of graphs with maximum degree at least 5.