Adjacent Vertex Distinguishing E-total Coloring Of Cartesian Products And Compositions Of Graphs

Let G=(V, E) be a simple, undirected and finite graph.A k-total coloring of a graph G is an assignment of k colors,1,2,…,k, to the vertices and edges of G. Let f be a k-total coloring of a graph G, and no two adja-cent vertices receive the same color, no vertex receives the same color as any one of its incident edges does, then f is called a E-total coloring of G using k colors. Given such a coloring f, for any vertex u∈V(G), let C(u) be the set of colors assigned to vertex u and edges incident to vertex u, if u∈V(G), u∈V(G), u is incident with u in G and C(u)≠C(u), then we say that f is a adjacent vertex distinguishing E-total coloring of graph G using k colors (in brief k - AVDETC). The minimum number of colors required for a AVDETC of G, denoted by xate(G), is called the adjacent vertex distinguishing E-total chromatic number. In Chapter 2, by using of the analysis method and the method of constructing concrete coloring, the ad-jacent vertex distinguishing E-total chromatic numbers for the cartesian products of the graphs with chromatic number at least 2 or the graphs with adjacent vertex distinguishing E-total chromatic numbers 3 or 4 are discussed. In Chapter 3, by using the method of combinational analysis and the method of constructing con-crete coloring, the adjacent vertex distinguishing E-total colorings of compositions of two graphs that satisfying several conditions are studied, and adjacent vertex distinguishing E-total chromatic numbers for compositions of several graphs, such as paths, cycles, fans and wheels, are obtained.