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

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 ∆(G) + 2p − 2 provided that ∆(G) ≥ ∆ and 1(G) ≥ 1, where (∆;1) ∈ {(6p +2;3);(4p +2;4);(2p +5;5)}. As a consequence, we generalize andimprovesomeresultsobtainedin(F.Bazzaro, M.Montassier, A.Raspaud, (d;1)-Totallabellingofplanar graphs with large girth and high maximum degree, Discrete Math. 307 (2007) 2141-2151).