LAUSR

A k-(p, 1)-total labelling of a graph G is a function f from $$V(G)\cup E(G)$$V(G)∪E(G) to the color set $$\{0, 1, \ldots , k\}$${0,1,…,k} such that $$|f(u)-f(v)|\ge 1$$|f(u)-f(v)|≥1 if $$uv\in E(G), |f(e_1)-f(e_2)|\ge 1$$uv∈E(G),|f(e1)-f(e2)|≥1 if $$e_1$$e1 and $$e_2$$e2 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. The minimum k such that G has a k-(p, 1)-total labelling, denoted by $$\lambda _p^T(G)$$λpT(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 $$\Delta \ge 4p+4$$Δ≥4p+4 and $$p\ge 2, \lambda _p^T(G)\le \Delta +2p-2$$p≥2,λpT(G)≤Δ+2p-2.