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Neighbor sum distinguishing list total coloring of IC-planar graphs without 5-cycles

Donghan Zhang — 2022

Czechoslovak Mathematical Journal

Let $G = (V (G), E (G))$ be a simple graph and $E_{G} (v)$ denote the set of edges incident with a vertex $v$ . A neighbor sum distinguishing (NSD) total coloring $φ$ of $G$ is a proper total coloring of $G$ such that $\sum_{z \in E_{G} (u) \cup {u}} φ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} φ (z)$ for each edge $u v \in E (G)$ . Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $Δ$ admits an NSD total $(Δ + 3)$ -coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $Δ \geq 11$ but without $5$ -cycles by applying the Combinatorial Nullstellensatz.

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