# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## 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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