The crossing number of the complete 4-partite graph $K_{1,1,m,n}$

TL;DR

This paper establishes new lower bounds for the crossing number of the complete 4-partite graph $K_{1,1,m,n}$, linking these bounds to the well-known Zarankiewicz's and Harborth's conjectures, and confirming the conjecture under certain conditions.

Contribution

The paper derives explicit lower bounds for $ extrm{cr}(K_{1,1,m,n})$ based on the parity of $m$ and $n$, connecting these bounds to existing conjectures and verifying the Harborth conjecture in specific cases.

Findings

Lower bounds depend on the parity of m and n.

If both m and n are even and ZC is true, HC holds for $K_{1,1,m,n}$.

If at least one of m or n is odd and ZC and HC on $K_{2,m,n}$ are true, then HC holds for $K_{1,1,m,n}$.

Abstract

Let $\textrm{cr}(G)$ denote the crossing number of a graph $G$. The well-known Zarankiewicz's conjecture (ZC) asserted $\textrm{cr}(K_{m,n})$ in 1954. In 1971, Harborth gave a conjecture (HC) on $\textrm{cr}(K_{x_1,...,x_n})$. HC on $K_{1,m,n}$ is verified if ZC is true by Ho et al. in 2021. In this paper, we showed the following results: If both $m$ and $n$ are even, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{m+1,n+3})+\textrm{cr}(K_{m+3,n+1})-mn-\frac{1}{4}(m^2+n^2));\] If both $m$ and $n$ are odd, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{1,m+1,n+1})+\textrm{cr}(K_{2,m,n})-\frac{1}{4}(m+1)(n+1)+1);\] If $m$ is even and $n$ is odd, then \begin{equation}\nonumber \begin{split} \textrm{cr}(K_{1,1,m,n})&\geq \frac{1}{4}(\textrm{cr}(K_{m+1,n+2})+\textrm{cr}(K_{m+3,n+2})+2\textrm{cr}(K_{2,m,n}) \\&-m(n+1)-\frac{1}{4}(n+1)^2). \end{split} \end{equation} The lower bounds in our result imply that if both $m$ and $n$ are even and ZC is true, then HC on $K_{1,1,m,n}$ holds; if at least one of $m$ and $n$ is odd and both ZC and HC on $K_{2,m,n}$ are true, then HC on $K_{1,1,m,n}$ holds.