Study On Several Dominating Coloring Parameters Of Graphs

Graph theory is a branch of discrete mathematics which is widely used and rich in content.The coloring theory of graph originated from the famous four color conjecture.It is a very important research topic in graph theory,and has important applications in combinatorial optimization,coding calculation and interactive network.Scholars' in depth study of coloring has greatly promoted the development of graph theory.In recent years,various generalizations of proper coloring of graphs have been put forward one after another,such as dominator coloring,domination coloring,dominated coloring,total dominator coloring,total domination coloring,total dominator edge coloring,and more.Graph coloring theory and graph control set theory are two very important research topics in graph theory.It is widely used in many fields such as computer science.Although the research results of controlled coloring are few,the application of dominated coloring in genetic network has been found.This paper mainly discusses dominating colorings problems for graphs,including domination coloring,total dominator coloring and total domination coloring of graphs.About the domination coloring of graphs:through the total control number ?t(G)and the independent number ?0(G)of the connected graph G,get the G of the upper or lower bound of the domination chromatic number.For a tree T with a fixed number of supporting vertices,the lower bound of ?dd(T)is obtained by studying the structure of the tree T.The value of the domination chromatic number of the Mycielskian graph M(G)of the general graph G is calculated by the method of proof by contradiction,and the generalized Mycielskian graph Mk(G)of the general graph G,is obtained the upper and lower bounds of the domination chromatic number according to the structure of the graph:?dd(G)+k??dd(Mk(G)??dd(G)+1+kn.About the total dominator coloring of graphs:we determine the upper and lower bounds of the total dominator chromatic number of the middle graph of the general graph G by the structural properties of the graph:[n/2]+1??dt(M*(G))?n+[t/2].By analyzing the structural properties of the Middle graph and Total graph,Middle graph M*(Kn),M*(Kn,m)and M*(Sn,m)of the complete graph Kn,complete bipartite graph Kn,m and double star graphs Sn,m are calculated values of the total dominator chromatic number and obtained exact value of the total dominator chromatic number of the Total graph T(Kn)and T(Sn,m)of double star graph Sn,m and complete graph Kn.About the total domination coloring of graphs:through the transformation and structural comparison of the points or edges of the graph,the relationship between the total domination chromatic number of the general graph G and the transformed graph G is found.For general graphs G and integers k?1,it is proved by graph construction that ?td(G)=k is NP-complete.