Zarankiewiczʼs Conjecture is finite for each fixed m

Abstract

Zarankiewiczʼs Crossing Number Conjecture states that the crossing number $\mathrm{cr}(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ equals $Z(m,n):=\lfloor m/2\rfloor \lfloor (m-1)/2\rfloor \lfloor n/2\rfloor \lfloor (n-1)/2\rfloor$, for all positive integers m, n. This conjecture has only been verified for $\min \{m,n\}⩽6$, for $K_{7,7}$, $K_{7,8}$, $K_{7,9}$, and $K_{7,10}$, and for $K_{8,8}$, $K_{8,9}$, and $K_{8,10}$. We determine, for each positive integer m, an integer $N_0=N_0(m)$ with the following property: if $\mathrm{cr}(K_{m,n})=Z(m,n)$ for all $n⩽N_0$, then $\mathrm{cr}(K_{m,n})=Z(m,n)$ for every n. This yields, for each fixed integer m, a finite algorithm that either shows that the assertion: “for all n, $\mathrm{cr}(K_{m,n})=Z(m,n)$” is true, or else finds a counterexample.