Light edges in 1-planar graphs of minimum degree 3

Abstract

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of type $(3,\le 23)$ or $(4,\le 11)$ or $(5,\le 9)$ or $(6,\le 8)$ or $(7,7)$. Moreover, the upper bounds $9,8$ and $7$ here are sharp and the upper bounds $23$ and $11$ are very close to the possible sharp ones, which may be 20 and 10, respectively. This generalizes a result of Fabrici and Madaras [Discrete Math., 307 (2007) 854--865] which says that each 3-connected 1-planar graph contains a light edge, and improves a result of Hudák and Šugerek [Discuss. Math. Graph Theory, 32(3) (2012) 545--556], which states that each 1-planar graph of minimum degree at least $4$ contains an edge with degrees of its endvertices of type $(4,\le 13)$ or $(5,\le 9)$ or $(6,\le 8)$ or $(7,7)$.

This paper was submitted to Discrete Mathematics on Dec.4, 2018, and will be published there