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.

Zarankiewicz的猜想指出，完整二分图\（K_ {m，n} \）的交叉数\（\ text {cr}（K_ {m，n}）\）是\（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 \），其中\（\ lfloor x \ rfloor \）表示不超过x的最大整数。据推测，完整三方图\（K_ {1，m，n} \）的交叉数\（\ text {cr}（K_ {1，m，n}）\）为\（Z（m + 1，n + 1）-\ lfloor \ frac {m} {2} \ rfloor \ lfloor \ frac {n} {2} \ rfloor \）。当m和n之一甚至是偶数，Ho在2008年证明了Zarankiewicz的猜想成立时，这个猜想是成立的。当m和n都为奇数时，Ho证明了\（\ 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 \），并猜想在这种不平等中存在平等。哪一个猜想是正确的？在本文中，我们证明了如果Zarankiewicz的猜想成立，那么前者是正确的。