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

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 $$uv\in E(G), |f(e_1)-f(e_2)|\ge 1$$ u v ∈ E ( G ) , | f ( e 1 ) - f ( e 2 ) | ≥ 1 if $$e_1$$ e 1 and $$e_2$$ e 2 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 $$\lambda _p^T(G)$$ λ 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 $$\Delta \ge 4p+4$$ Δ ≥ 4 p + 4 and $$p\ge 2, \lambda _p^T(G)\le \Delta +2p-2$$ p ≥ 2 , λ p T ( G ) ≤ Δ + 2 p - 2 .