Geodesic
Light edges in 1-planar graphs with prescribed minimum degree
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 545-556

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A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains just the selected edge type).
Keywords: light edge, 1-planar graph 
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Hudák, Dávid; Šugerek, Peter. Light edges in 1-planar graphs with prescribed minimum degree. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 545-556. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a12/