Geodesic
On (p, 1)-Total Labelling of Some 1-Planar Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 531-558

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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. In this paper, it is proved that the (p, 1)-total labelling number (p ≥ 2) of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ (G) ≥ 6p + 7 or Δ (G) ≥ 4p + 6 and G is triangle-free.
Keywords: 1-planar graph, total coloring, structural theorem, (p, 1)-total labelling 
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     author = {Niu, Bei and Zhang, Xin},
     title = {On (p, {1)-Total} {Labelling} of {Some} {1-Planar} {Graphs}},
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Niu, Bei; Zhang, Xin. On (p, 1)-Total Labelling of Some 1-Planar Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 531-558. http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a12/