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Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph G has a drawing in the plane so that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, it is proved that the (p, 1)-total labelling number of every IC-planar graph G is at most Δ(G) + 2p - 2 provided that Δ(G) ≥ Δ and g(G) ≥ g, where (Δ; g) ∈ {(6p+2; 3); (4p+2; 4); (2p+5; 5)}. As a consequence, we generalize and improve some results obtained in [F. Bazzaro, M. Montassier, A. Raspaud, (d, 1)-Total labelling of planar graphs with large girth and high maximum degree, Discrete Math. 307 (2007) 2141-2151].

On (p,1)-total labelling of plane graphs with independent crossings