Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

Let be a graph and be a -total coloring. Let denote the sum of color on a vertex and colors assigned to edges incident to . If whenever , then is called a neighbor sum distinguishing total coloring. The smallest integer such that has a neighbor sum distinguishing -total coloring is denoted by . In 2014, Dong and Wang obtained the results about depending on the value of maximum average degree. A -assignment of is a list assignment of integers to vertices and edges with for each vertex and for each edge . A total--coloring is a total coloring of such that whenever and whenever . We state that has a neighbor sum distinguishing total--coloring if has a total--coloring such that for all . The smallest integer such that has a neighbor sum distinguishing total--coloring for every -assignment is denoted by . In this paper, we strengthen results by Dong and Wang by giving analogous results for .

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