Proper Additive Colorings Of Some Classes Of Graphs

A proper additive coloring(or proper lucky labeling)of a graphis an assignment of positive integers{1,2,···,6)}to the vertices ofsuch that no adjacent vertices ofreceive the same number,and for every two adjacent vertices the sums of numbers assigned to their neighbors are different.The minimum number6)for which there exists a proper additive coloring ofis called proper additive chromatic number(or proper lucky number),and is denoted by().In this paper,we investigate the proper additive coloring of some basic simple graphs,special regular multipartite graphs and their operation graphs,and determine the corresponding chromatic numbers or its tight upper bound.The main work of this paper includes the following four parts.(1)We introduce the concepts of uniformly regular multipartite graphs and same type uniformly regular multipartite graphs,and obtain a necessary condition for the judgment of uniformly regular multipartite graphs.(2)We give the precise value of the proper additive chromatic number of basic simple graphs such as paths,cycles,fans,wheels,etc.And we obtain the tight upper bound of the proper additive chromatic number of a uniformly regular multipartite graphs,and based on this,get the corresponding chromatic numbers of regular bipartite graphs,complete bipartite graphs,and balanced complete multipartite graphs.(3)We analyze the proper additive coloring of the unary operation graphs of some special graphs,and determine the corresponding chromatic numbers or its tight upper bound,involving unary operation graphs mainly includes:the complement of a complete bipartite graph,the complement of a special regular bipartite graph,the square graph of a path,the double graph of any nontrivial graph,the Mycielski graph of a uniformly regular multipartite graph,etc.(4)We analyze the proper additive coloring of the binary operation graphs of some special graphs,and determine the corresponding chromatic numbers or its tight upper bound,involving binary operation graphs mainly includes:the Cartesian product of uni-formly regular multipartite graphs,the join of same type uniformly regular multipartite graphs,the lexicographic product of uniformly regular multipartite graphs,and the corona product of same type uniformly regular multipartite graphs,etc.