On the lightness of chordal 4-cycle in 1-planar graphs with high minimum degree

Abstract

A graph G is 1 -planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. The family of 1-planar graphs with minimum vertex degree at least δ and minimum edge degree at least ɛ is denoted by P δ 1 ( ɛ ) . In this paper, it is proved that every graph in P 7 1 (14) (resp. P 6 1 (13) ) contains a copy of chordal 4-cycle with all vertices of degree at most 10 (resp. 12).