Neighbor sum distinguishing total chromatic number of planar graphs without 4-cycles

Abstract

Let ø be a proper k-total coloring of G = (V(G), E(G)) by using the color set {1,2,..., k}. For any v ϵ V(G), let f(v) = Σ_uvϵE(G) ø(uv) + ø(v). The coloring ø is neighbor sum distinguishing if f(u) = f(v) for each edge uv ϵ E(G). The neighbor sum distinguishing total chromatic number is the smallest number k in such a coloring of G and denoted by Xϵ(G). In this paper, we determine Xϵ(G) for any planar graph G without 4-cycles and δ(G) ≥ 9. © 2017 Utilitas Mathematica Publishing Inc.. All rights reserved.