The Crossing Number of \(K_{1,1,3,n}\)

Abstract

In this paper, we show that the crossing number of the complete multipartite graph \(K_{1,1,3,n}\) is

\[\operatorname{cr}(K_{1,1,3,n}) = 4\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor + \lfloor\frac{3n}{2}\rfloor\]

Our proof depends on Kleitman’s results for the complete bipartite graphs [D. J. Kleitman, The crossing number of \(K_{5,n}\), J. Combin.Theory, \(9 (1970), 315-323\)]..