Bull. Korean Math. Soc. 2017; 54(3): 763-787
Online first article November 3, 2016 Printed May 31, 2017
https://doi.org/10.4134/BKMS.b160296
Copyright © The Korean Mathematical Society.
Zai Ping Lu and Ning Song
Nankai University, Nankai University
A graph is called \emph{$1$-planar} if it can be drawn in the Euclidean plane $\mathbb{R}^2$ such that each edge is crossed by at most one other edge. The \emph{weight} of an edge is the sum of degrees of two ends. It is known that every planar graph of minimum degree $\delta\ge3$ has an edge with weight at most $13$. In the present paper, we show the existence of edges with weight at most $25$ in $3$-connected $1$-planar graphs.
Keywords: 1-planar graph, weight, light edge
MSC numbers: 05C10, 68R10
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