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

图G的A ( p , 1)-总标记是映射f : \(V(G)\cup E(G)\) \(\rightarrow \) \(\{0, 1, \cdots , k \}\)使得\(|f(u)-f(v)|\ge 1\)如果\(uv\in E(G)\)，\(|f(e_1)-f(e_2)| \ge 1\)如果\(e_1\)和\(e_2\)是G中的两条相邻边，并且\(|f(u)-f(e)|\ge p\)如果顶点u与边缘e . 在本文中，我们重点关注 a ( p , 1)-总标签的列表版本。给定一个族\(L=\{L(u)\subseteq \mathbb {N}:u\in V(G)\cup E(G)\}\)，一个L -list ( p , 1)-total G的标签是G的a ( p , 1)-总标签f，使得每个元素\(u\in V(G) \cup E(G) 的 \(f(u)\in L(u)\) \)。如果当族L仅包含大小至少为k 的集合时，如果图G承认L列表 ( p , 1)-total 标记，则称图 G是 ( p , 1)- k -total 可选择的。图G的 ( p , 1)- k -total 可选择的最小k是G的列表 ( p , 1)-total 标记数，表示为\(\lambda _{lp}^T(G)\ ）。在本文中，我们首先使用与组合零值相关的一些重要定理来证明循环\ ( C_n\)的\(\lambda _{lp}^T(C_n)\)的上界为\(2p+1\) )与\(p\ge 2\)。设G为最大度数\(\Delta (G)\ge 6p+3\)的图。然后我们证明如果G是平面图或没有相邻 3 圈的 1 平面图，则\(\lambda _{lp}^T(G)\le \Delta (G)+2p-1\) ( \(p\ge 2\) )。