Discussiones Mathematicae Graph Theory 32(3) (2012) 545-556
doi: 10.7151/dmgt.1625

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Light Edges in 1-planar Graphs with Prescribed Minimum Degree

Dávid Hudák and Peter Šugerek

Institute of Mathematics, Faculty of Science,
Pavol Jozef Šafárik University,
Jesenná 5, 040 01 Košice, Slovakia

Abstract

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

2010 Mathematics Subject Classification: 05C10.

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Received 3 March 2011
Revised 29 September 2011
Accepted 29 September 2011