Light subgraphs in the family of 1-planar graphs with high minimum degree

A graph is 1-planar if it can be drawn in 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 with minimum degree 7 contains a copy of K 2.(K 1.K 2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K 1.(K 1.K 2) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.