Abstract
A (p, 1)-total labelling of a graph G is a mapping f: \(V(G)\cup E(G)\) \(\rightarrow \) \(\{0, 1, \cdots , k\}\) such that \(|f(u)-f(v)|\ge 1\) if \(uv\in E(G)\), \(|f(e_1)-f(e_2)|\ge 1\) if \(e_1\) and \(e_2\) are two adjacent edges in G and \(|f(u)-f(e)|\ge p\) if the vertex u is incident with the edge e. In this paper, we focus on the list version of a (p, 1)-total labelling. Given a family \(L=\{L(u)\subseteq \mathbb {N}:u\in V(G)\cup E(G)\}\), an L-list (p, 1)-total labelling of G is a (p, 1)-total labelling f of G such that \(f(u)\in L(u)\) for every element \(u\in V(G)\cup E(G)\). A graph G is said to be (p, 1)-k-total choosable if it admits an L-list (p, 1)-total labelling whenever the family L contains only sets of size at least k. The smallest k for which a graph G is (p, 1)-k-total choosable is the list (p, 1)-total labelling number of G, denoted by \(\lambda _{lp}^T(G)\). In this paper, we firstly use some important theorems related to Combinatorial Nullstellensatz to prove that the upper bound of \(\lambda _{lp}^T(C_n)\) for cycles \(C_n\) is \(2p+1\) with \(p\ge 2\). Let G be a graph with maximum degree \(\Delta (G)\ge 6p+3\). Then we prove that if G is a planar graph or a 1-planar graph without adjacent 3-cycles, then \(\lambda _{lp}^T(G)\le \Delta (G)+2p-1\) (\(p\ge 2\)).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 12101285, 12171222, 11771376, 12071411, 11571252), Guangdong Philosophy and Social Sciences Planning Project (Grant No. GD22CXW01), Research Platforms and Projects of Colleges and Universities in Guangdong (Grant No. 2022KTSCX071), Guangdong basic and applied basic research foundation (Natural Science Foundation of Guangdong Province), China (No. 2021A1515010254), Foundation of Lingnan Normal University (ZL2021017, ZL1923).
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This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 12101285, 12171222).
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Sun, L., Yu, G. & Wu, J. On list (p, 1)-total labellings of special planar graphs and 1-planar graphs. J Comb Optim 47, 15 (2024). https://doi.org/10.1007/s10878-024-01111-3
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DOI: https://doi.org/10.1007/s10878-024-01111-3