The Conjecture on the Crossing Number of $K_{1,m,n}$ is true if Zarankiewicz’s Conjecture Holds

Abstract

Zarankiewicz’s conjecture states that the crossing number $\text {cr}(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ is $Z(m,n):=\lfloor \frac{m}{2}\rfloor \lfloor \frac{m-1}{2}\rfloor \lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor$, where $\lfloor x \rfloor$ denotes the largest integer no more than x. It is conjectured that the crossing number $\text {cr}(K_{1,m,n})$ of the complete tripartite graph $K_{1,m,n}$ is $Z(m+1,n+1)-\lfloor \frac{m}{2}\rfloor \lfloor \frac{n}{2}\rfloor$. When one of m and n is even, Ho proved that this conjecture is true if Zarankiewicz’s conjecture holds, in 2008. When both m and n are odd, Ho proved that $\text {cr}(K_{1,m,n})\ge \text {cr}(K_{m+1,n+1})-\left\lfloor \frac{n}{m}\lfloor \frac{m}{2}\rfloor \lfloor \frac{m+1}{2}\rfloor \right\rfloor$ and conjectured that equality holds in this inequality. Which one of the conjectures may be true? In this paper, we proved that if Zarankiewicz’s conjecture holds, then the former one is true.