Bulletin of the
Korean Mathematical Society BKMS

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