Triple crossing numbers of graphs

Abstract

We introduce the triple crossing number, a variation of crossing number, of a graph, which is the minimal number of crossing points in all drawings with only triple crossings of the graph. It is defined to be zero for a planar graph, and to be infinite unless a graph admits a drawing with only triple crossings. In this paper, we determine the triple crossing numbers for all complete multipartite graphs including all complete graphs.Comment: 34 pages, 53 figures: We reorganized the article and revised some argument