On "r , S , T" Coloring Of Some Special Graphs

The coloring of the graphs has been one of the main problems. It has important theoretical and practical value. In recent years, coloring problem in graph theory is very active, producing many new problems and new coloring methods, get a series of excellent results. After continues the vertex coloring, edge coloring and total coloring, in 2007, A. Kemnitz and M. Marangio put forward the concept of [r,s,t]-coloring of graphs, it is a generalization of classical vertex coloring,edge coloring and total coloring of a graph, which has broad significant applications . This paper discusses the [r, s, t]- coloring of some special graphs.Firstly, we introduce some basic concepts,background and purposes. We summarize the current reseach on the vertex coloring, edge coloring and total coloring and related basic concepts, the study methods of them can be used in the [r,s,t]-coloring. At the same time, the introduction of A. Kemnitz and M. Marangio proposed the concept of [r,s,t]-coloring, and reviewed the current research status. A. Kemnitz and M. Marangio gives the [r,s,t]-chromatic number of general graphs of the boundary and some related properties, also discussed the [r,s,t]-chromatic number of the complete graph of order n. According to A. Kemnitz's some properties of the [r,s,t]-chromatic number, we get several parallel conclusions.Secondly, the special graphs have some special structure and characteristics, so many researches of the problems of the graph theory are proceeded from them, so that people can obtain the general results.And so it is the coloring problem.Therefore, chapter 3 studies the [r,s,t]-chromatic number of some special graphs, such as the windmill graph K₃ⁿ, graph D_m,4,gears graph (W|⁾_n. we first get the chromatic number,chromatic index and total chromatic number of these special graphs, then according to the general bound of the [r,s,t]- chromatic number and some properties, we discuss, when the parameter r, s, t meet certain conditions, the [r,s,t]- chromatic number of special graphs.Finally, for researching the join of two disjoint graphs is a common method, so in Chapter 4, we discuss the empty graph and circle, circle and circle, these two kinds of join graph of the [r,s,t]- chromatic number. At first , we precent two kinds of join graphs chromatic number and chromatic index . we get , if the circle order equal empty graph order plus one,then it is class 2 , the rest is class one. The join of circle and circe if one circle order is more or less one than the other circle order , then it is class 2 , the rest is class one. then according to the general bound of the [r,s,t]- chromatic number and some properties, we discuss, when the parameter r, s and t meet certain conditions, the [r,s,t]- chromatic number of these two kinds of jion graph.