Neighbor Sum Distinguishing Total Choosability of Cubic Graphs

Let G = (V, E) be a graph and R be the set of real numbers. For a k-list total assignment L of G that assigns to each member z is an element of V boolean OR E a set L-z of k real numbers, a neighbor sum distinguishing (NSD) total L-coloring of G is a mapping phi : V boolean OR E -> R such that every member z is an element of V boolean OR E receives a color of L-z, every pair of adjacent or incident members in V boolean OR E receive different colors, and Sigma(z is an element of EG(u)boolean OR{u}) phi(z) not equal Sigma(z is an element of EG(v)boolean OR{v}) phi(z) for each edge uv is an element of E, where E-G(v) is the set of edges incident with v in G. In 2015, Pilsniak and Wozniak posed the conjecture that every graph G with maximum degree Delta has an NSD total L-coloring with L-z = {1, 2, ... , Delta + 3} for any z is an element of V boolean OR E, and confirmed the conjecture for all cubic graphs. In this paper, we extend their result by proving that every cubic graph has an NSD total L-coloring for any 6-list total assignment L.