Research On Vertex(edge) Dominating Colorings Of Graphs

With the development of computer network,domination theory of graph has been extensively studied and has rapidly developed into an important research field in graph theory.At the same time,as the key and hot research in graph theory,the coloring theory of graphs is also developing,and various coloring problems with constraints are raised.Based on the study of domination set and vertex coloring problems,in the coloring problem of graphs,Gera et al.proposed the new research parameter of graph's dominator coloring.Since then,new colorings combined with domination and coloring have been put forward by scholars,such as(total)dominator coloring,domination coloring,dominated coloring and so on.At present,the research on some dominating colorings is not in-depth,but it has been determined that some dominating colorings have specific practical significance in social networks and genetic networks.Therefore,the study of some dominating colorings is necessary and meaningful.For the dominating colorings problem,most scholars have studied the bounds or values of dominating chromatic numbers of graphs,and the complexity of dominating colorings;characterize graphs that satisfy some conditions.However,the results of the dominating edge colorings problem of graphs are relatively few.In this paper,we first study dominator edge coloring and the stability(bondage)number of total dominator edge chromatic numbers.The dominator edge coloring of graphs is generalized by dominator coloring and total dominator edge coloring.The stability(bondage)num-ber of total dominator edge chromatic numbers are extended by the stability(bondage)number of total dominator chromatic numbers.In this paper,we mainly study the vertex(edge)dominating coloring of graphs.The main research method is to construct a conditional coloring scheme according to the structure of graph;proving the feasibility of the coloring scheme;and according to the conclusion,judging whether it is the optimal coloring scheme.The research content includes three parts:dominator edge coloring of graphs,total dominator edge coloring of graphs,and dominated coloring of product graphs.On the research of the dominator edge coloring of graphs,the bounds of the dominator edge chromatic num-ber of general graphs are given:max{?^?(G),?^?(G)}??_d^?(G)??^?(G)+?^?(G);the values of the dominator edge chromatic number of special graphs such as path-s,cycles and complete graphs are obtained;the relationship of the dominator edge chromatic number between the graph and operational graphs is analyzed.On the re-search of the total dominator edge coloring of graphs,the bounds of the total domina-tor edge chromatic number of general graphs are given;the values of total dominator edge chromatic numbers of special graphs such as path,cycle and star graph are ob-tained;the stability number and bondage number of total dominator edge chromatic numbers of special graphs are calculated respectively.On the research of the dom-inated coloring of product graphs,the value of the dominated chromatic number of direct product graphs in complete p-partite graph and complete q-partite graph is giv-en:?_dom(G×H)=p+2(q?p?2),the bounds or values of the dominated chromatic numbers of Cartesian product and lexicographic product graphs constructed by some graphs are given respectively.