On (p, 1)-total labelling of planar graphs

Abstract

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 $p\ge 2, \lambda _p^T(G)\le \Delta +2p-2$.