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

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. A k-(p, 1)-total labelling of a graph G is a function f from V(G) boolean OR E(G) to the color set {0,1, ..., k} such that vertical bar f(u) - f(v)vertical bar >= 1 if uv epsilon E(G), vertical bar f(e(1)) - f(e(2))vertical bar >= 1 if e(1) and e(2) are two adjacent edges in G and vertical bar f(u) - f(e)vertical bar >= p if the vertex u is incident to the edge e. The minimum kappa such that G has a k-(p, 1)-total labelling, denoted by lambda(T)(p)(G), is called the (p, 1)-total labelling number of G. In this paper, we prove that, if a 1-planar graph G satisfies that maximum degree Delta(G) >= 7p + 1 and no adjacent triangles in G or maximum degree Delta(G) >= 6p + 3 and no intersecting triangles in G, then lambda(T)(p)(G) <= Delta + 2p - 2, p >= 2.