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

A (p, 1)-total labelling of a graph G is a mapping f: $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) $$\rightarrow $$ → $$\{0, 1, \cdots , k\}$$ { 0 , 1 , ⋯ , k } such that $$|f(u)-f(v)|\ge 1$$ | f ( u ) - f ( v ) | ≥ 1 if $$uv\in E(G)$$ u v ∈ E ( G ) , $$|f(e_1)-f(e_2)|\ge 1$$ | f ( e 1 ) - f ( e 2 ) | ≥ 1 if $$e_1$$ e 1 and $$e_2$$ e 2 are two adjacent edges in G and $$|f(u)-f(e)|\ge p$$ | f ( u ) - f ( e ) | ≥ 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 $$L=\{L(u)\subseteq \mathbb {N}:u\in V(G)\cup E(G)\}$$ L = { L ( u ) ⊆ N : u ∈ V ( G ) ∪ E ( G ) } , an L-list (p, 1)-total labelling of G is a (p, 1)-total labelling f of G such that $$f(u)\in L(u)$$ f ( u ) ∈ L ( u ) for every element $$u\in V(G)\cup E(G)$$ u ∈ V ( G ) ∪ 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 $$\lambda _{lp}^T(G)$$ λ lp T ( G ) . In this paper, we firstly use some important theorems related to Combinatorial Nullstellensatz to prove that the upper bound of $$\lambda _{lp}^T(C_n)$$ λ lp T ( C n ) for cycles $$C_n$$ C n is $$2p+1$$ 2 p + 1 with $$p\ge 2$$ p ≥ 2 . Let G be a graph with maximum degree $$\Delta (G)\ge 6p+3$$ Δ ( G ) ≥ 6 p + 3 . Then we prove that if G is a planar graph or a 1-planar graph without adjacent 3-cycles, then $$\lambda _{lp}^T(G)\le \Delta (G)+2p-1$$ λ lp T ( G ) ≤ Δ ( G ) + 2 p - 1 ( $$p\ge 2$$ p ≥ 2 ).