On local properties of 1-planar graphs with high minimum degree
David E. Hudak, Tomáš Madaras · Ars Mathematica Contemporanea · 2011
A graph is called 1-planar if there exists its drawing in the plane such that each edge contains at most one crossing. We prove that each 1-planar graph of minimum degree 7 contains a pair of adjacent vertices of degree 7 as well as several small graphs whose vertices have small degrees; we also prove the existence of a 4-cycle with relatively small degree vertices in 1-planar graphs of minimum degree at least 6.