Drawing graph joins in the plane with restrictions on crossings

Abstract

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. In 2014, Zhang showed that the set of all 1-planar graphs can be decomposed into three classes C0,C1 and C2 with respect to the types of crossings. He proved that every n-vertex 1-planar graph of class C1 has a C1-drawing with at most 3/5n-6/5 crossings. Consequently, every n-vertex 1-planar graph of class C1 has at most 18/5n ? 36/5 edges. In this paper we prove a stronger result. We show that every C1-drawing of a 1-planar graph has at most 3/5n ? 6/5 crossings. Next we present a construction of n-vertex 1-planar graphs of class C1 with 18/5n ? 36/5 edges. Finally, we present the decomposition of 1-planar join products.