Let be a graph and : ( ) ∪ ( ) → {1, 2, 3, . . . , } be a -total coloring. Let (V) denote the sum of color on a vertex V and colors assigned to edges incident to V. If ( ) ̸ = (V) whenever V ∈ ( ), then is called a neighbor sum distinguishing total coloring. The smallest integer such that has a neighbor sum distinguishing -total coloring is denoted by tndi ∑ ( ). In 2014, Dong and Wang obtained the results about tndi ∑ ( ) depending on the value of maximum average degree. A -assignment of is a list assignment of integers to vertices and edges with | (V)| = for each vertex V and | ( )| = for each edge . A total--coloring is a total coloring of such that (V) ∈ (V) whenever V ∈ ( ) and ( ) ∈ ( ) whenever ∈ ( ). We state that has a neighbor sum distinguishing total--coloring if has a total--coloring such that ( ) ̸ = (V) for all V ∈ ( ). The smallest integer such that has a neighbor sum distinguishing total--coloring for every -assignment is denoted by Ch ∑ ( ). In this paper, we strengthen results by Dong and Wang by giving analogous results for Ch ∑ ( ).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.