Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10

Abstract

A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that \(\sum\limits_{z \in {E_G}(u) \cup \{u\}} {\phi (z) \ne} \sum\limits_{z \in {E_G}(v) \cup \{v\}} {\phi (z)} \) for each edge uv ∈ E(G), where EG(u) is the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak conjectured that every graph with maximum degree Δ has an NSD total (Δ + 3)-coloring. Recently, Yang et al. proved that the conjecture holds for planar graphs with Δ ≥ 10, and Qu et al. proved that the list version of the conjecture also holds for planar graphs with Δ ≥ 13. In this paper, we improve their results and prove that the list version of the conjecture holds for planar graphs with Δ ≥ 10.