文摘
A k-(p, 1)-total labelling of a graph G is a function f from \(V(G)\cup E(G)\) to the color set \(\{0, 1, \ldots , k\}\) 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. The minimum k such that G has a k-(p, 1)-total labelling, denoted by \(\lambda _p^T(G)\), is called the (p, 1)-total labelling number of G. In this paper, we prove that, for any planar graph G with maximum degree \(\Delta \ge 4p+4\) and \(p\ge 2, \lambda _p^T(G)\le \Delta +2p-2\).