Neighbor Sum Distinguishing Total Choosability of Cubic Graphs

Abstract

Let $$G=(V, E)$$ be a graph and $${\mathbb {R}}$$ be the set of real numbers. For a k-list total assignment L of G that assigns to each member $$z\in V\cup E$$ a set $$L_{z}$$ of k real numbers, a neighbor sum distinguishing (NSD) total L-coloring of G is a mapping $$\phi :V\cup E \rightarrow {\mathbb {R}}$$ such that every member $$z\in V\cup E$$ receives a color of $$L_z$$ , every pair of adjacent or incident members in $$V\cup E$$ receive different colors, and $$\sum _{z\in E_{G}(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E_{G}(v)\cup \{v\}}\phi (z)$$ for each edge $$uv\in E$$ , 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 $$\Delta $$ has an NSD total L-coloring with $$L_z=\{1,2,\dots , \Delta +3\}$$ for any $$z\in V\cup E$$ , 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.