Neighbor sum distinguishing total choice number of IC-planar graphs with restrictive conditions

Abstract

A neighbor sum distinguishing (NSD) total coloring $ \phi $ of $ G $ is a proper total coloring such that $ \sum_{z\in E_{G}(u)\cup\{u\}}\phi(z)\neq\sum_{z\in E_{G}(v)\cup\{v\}}\phi(z) $ for each edge $ uv\in E(G) $. Pilśniak and Woźniak asserted that each graph with a maximum degree $ \Delta $ admits an NSD total $ (\Delta+3) $-coloring in 2015. In this paper, we prove that the list version of this conjecture holds for any IC-planar graph with $ \Delta\geq10 $ but without five cycles by applying the discharging method, which improves the result of Zhang (NSD list total coloring of IC-planar graphs without five cycles).

Keywords

• ic-planar graphs
• neighbor sum distinguishing total choice number
• combinatorial nullstellensatz
• discharging method