### New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree

*Xin Zhang, Jian-Liang Wu, Guizhen Liu*

#### Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge
is crossed by at most one other edge. In this paper, it is shown
that each 1-planar graph of minimum degree 6 contains a copy of
4-cycle with all vertices of degree at most 19. In addition, we also
show that the complete graph K

_{4}is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.Full Text: PDF