The Adjacent Vertex Distinguishing Acyclic Edge Coloring And Some Classes Of Improper Coloring

Introduce to the concepts of adjacent vertex-distinguishing acyclic edge coloring, ad-jacent vertex-distinguishing E-total coloring, adjacent vertex-distinguishing VE-total col-oring and adjacent vertex-distinguishing VI-total coloring and obtain some upper boundsof these chromatic numbers by the Lova′sz general Local lemma, respectively.Some preliminaries, such as basic concepts, theorems and lemmas which are incidentto probabilistic method are given in the frst part.The concept of the adjacent vertex-distinguishing acyclic edge coloring of graphs isintroduced, and the adjacent vertex-distinguishing acyclic edge chromatic numbers of somepower graphs and union graphs, for instance, the P_n^k、C_5n^k、W_n^k、F_n^k and P_m∨W_n、P_m∨F_n、P-m∨P_n、P_m∨S-n、C_m,nare given by using of the method of constructing concretecoloring in the second part.The adjacent vertex-distinguishing E-total coloring of graphs are discussed. An upperbound for the adjacent vertex-distinguishing E-total chromatic number of graphs is givenby probabilistic method in the third part.The adjacent vertex-distinguishing VE-total coloring of graphs are considered, andan upper bound for the adjacent vertex-distinguishing VE-total chromatic number of agraph is obtained by probabilistic method in the fourth part.The concept of adjacent vertex-distinguishing VI-total coloring of graphs is discussed,and the adjacent vertex-distinguishing VI-total chromatic numbers of some simple graphs,the path P_n, the cycle C_n, the complete graph K_n, the fan F_nand the wheel W_nare givenby using of the method of constructing concrete coloring. An upper bound for the adjacentvertex-distinguishing VI-total chromatic number of graphs is given by probabilistic methodin the ffth part.