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.
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The project is supported by the National Natural Science Foundation of China (No.12271438, No.12071370 and U1803263), the Science Found of Qinhai Province (No.2022-ZJ-753), Shaanxi Fundamental Science Research Project for Mathematics and Physics (No.22JSZ009), Shangluo University Doctoral Initiation Fund Project (No.22SKY112) and Shangluo University Key Disciplines Project (Discipline name: Mathematics)
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Zhang, Dh., Lu, Y., Zhang, Sg. et al. Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10. Acta Math. Appl. Sin. Engl. Ser. 40, 211–224 (2024). https://doi.org/10.1007/s10255-024-1110-y
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DOI: https://doi.org/10.1007/s10255-024-1110-y
Keywords
- planar graphs
- neighbor sum distinguishing total choosibility
- combinatorial nullstellensatz
- discharging method