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

Abstract

Let G be a graph and ϕ:V(G)∪E(G)→{1,2,3,…,k} be a k-total coloring. Let w(v) denote the sum of color on a vertex v and colors assigned to edges incident to v. If w(u)≠w(v) whenever uv∈E(G), then ϕ is called a neighbor sum distinguishing total coloring. The smallest integer k such that G has a neighbor sum distinguishing k-total coloring is denoted by tndi∑ (G). In 2014, Dong and Wang obtained the results about tndi∑ (G) depending on the value of maximum average degree. A k-assignment L of G is a list assignment L of integers to vertices and edges with L(v)=k for each vertex v and L(e)=k for each edge e. A total-L-coloring is a total coloring ϕ of G such that ϕ(v)∈L(v) whenever v∈V(G) and ϕ(e)∈L(e) whenever e∈E(G). We state that G has a neighbor sum distinguishing total-L-coloring if G has a total-L-coloring such that w(u)≠w(v) for all uv∈E(G). The smallest integer k such that G has a neighbor sum distinguishing total-L-coloring for every k-assignment L is denoted by Ch∑ ′′(G). In this paper, we strengthen results by Dong and Wang by giving analogous results for Ch∑ ′′(G).