Neighbor sum distinguishing list total coloring of subcubic graphs

Abstract

Let $G=(V, E)$ be a simple graph and denote the set of edges incident to a vertex v by E(v). The neighbor sum distinguishing (NSD) total choice number of G, denoted by $\mathrm{ch}_{\Sigma }^{t}(G)$, is the smallest integer k such that, after assigning each $z\in V\cup E$ a set L(z) of k real numbers, G has a total coloring $\phi$ satisfying $\phi (z)\in L(z)$ for each $z\in V\cup E$ and $\sum _{z\in E(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E(v)\cup \{v\}}\phi (z)$ for each $uv\in E$. In this paper, we propose some reducible configurations of NSD list total coloring for general graphs by applying the Combinatorial Nullstellensatz. As an application, we present that $\mathrm{ch}^{t}_{\Sigma }(G)\le \Delta (G)+3$ for every subcubic graph G.