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∈ V∪ E a set L_z of k real numbers, a neighbor sum distinguishing (NSD) total L-coloring of G is a mapping ϕ: V∪ E→ R such that every member z∈ V∪ E receives a color of L_z, every pair of adjacent or incident members in V∪ E receive different colors, and ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈ E, where E_G(v) is the set of edges incident with v in G. In 2015, Pilśniak and Woźniak posed the conjecture that every graph G with maximum degree Δ has an NSD total L-coloring with L_z= { 1 , 2 , ⋯ , Δ + 3 } for any z∈ V∪ 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.