On local structure of 1-planar graphs of minimum degree 5 and girth 4
David E. Hudak, Tomáš Madaras · Discussiones Mathematicae Graph Theory · 2009
A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K1,4 with all vertices having degree at most 11.