On list (p, 1)-total labellings of special planar graphs and 1-planar graphs

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 \(\lambda _{lp}^T(G)\le \Delta (G)+2p-1\) (\(p\ge 2\)).