On $(p,1)$-Total Labelling of Plane Graphs With Independent Crossings

Xin Zhang, Guizhen Liu, Yong Yu

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. It is proved that the $(p,1)$-total labelling number of every IC-planar graph $G$ is at most $\Delta(G)+2p-2 $ provided that $\Delta(G)\geq \Delta$ and $g(G)\geq g$, where $(\Delta,g)\in \{(6p+2,3),(4p+2,4),(2p+5,5)\}$. 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].