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 tndiΣ(G). In 2014, Dong and Wang obtained the results about tndiΣ(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 nv ∈ 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″Σ(G). In this paper, we strengthen results by Dong andWang by giving analogous results for Ch″Σ(G).