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