On Some Colorings Of Graph

An incidence coloring of G is a mappingσfrom I(G) to a color set C, such that any two adjacent incidences in I(G)have different images. Ifσ:I(G)â†'C is an incidence coloring of G and|C|=k, k is an positive integer, then we say that a is a k-incidence coloring of G, and the minimum value of k such that G is k-incidence colorable is called the incidence chromatic number of G, and is denoted byχi,(G), namelyχi (G)= min{k|there exists an k-incidence coloring of G}.The definition of adjacent vertex distinguishing incidence coloring is based on the incidence coloring, which further defines parameters of graph coloring. If for any uv∈E(G),C(u)≠C(v), then we callσa k-adjacent vertex distinguishing incidence coloring of G. Andχai(G)= min{k| there exists an k-adjacent vertex distinguishing incidence coloring of G} is the adjacent vertex distinguishing incidence chromatic number of G.In this dissertation, from the construction of graph, using the methods of Reduction to Absurdity and Enumeration, we have researched the adjacent vertex distinguishing incidence coloring of some graphs, then we shall define the concepts of the k-d-distance vertex distinguishing incidence coloring and vertex distinguishing incidence coloring, then discusse some k-d-distance vertex distinguishing incidence coloring of some graphs is as follows. At last, we shall define the concepts of the vertex distinguishing incidence coloring and the vertex-distinguishing incidence chromatic number of graphs, and discussing some vertex distinguishing incidence coloring of some graphs.There are six chapters in this dissertation. The arrangement is as follows:In the three, we mainly determined the adjacent vertex distinguishing incidence chromatic numbers ofθ-graphs, generalized star graphs and generalizedθ-graphs.In the chapter four, based on the concepts of the adjacent vertex distinguishing incidence coloring, we developed the concepts of the k-d-distance vertex distinguishing incidence coloring and the k-d-distance vertex distinguishing incidence chromatic number at first, then studied some properties of it and some k-d-distance vertex distinguishing incidence chromatic numbers of some graphs, determined the k-2-distance vertex distinguishing incidence chromatic numbers of star Sn, fan Fn+1(n≥3), wheel Wn+1(n≥4), circle Cn(n≥4) and sunflower graphs, and some k-d-distance vertex distinguishing incidence chromatic numbers of path Pn(n≥5). In the chapter five, based on the concepts of the adjacent vertex distinguishing incidence coloring and chapter four, we developed the concepts of the vertex distinguishing incidence coloring and the vertex distinguishing incidence chromatic number at first, then studied vertex distinguishing incidence chromatic numbers of some graphs, determined the vertex distinguishing incidence chromatic numbers of star Sn, couplet graph Kn, fan Fn+1 (n> 3), wheel Wn+1 (n≥4) and C5, and determined the lowerbounds of the vertex distinguishing incidence chromatic numbers of Pn(n≥5)and Cn(n≥4).