On the characterization of some algebraically defined bipartite graphs of girth eight

Abstract

For any field $\mathbb{F}$ and polynomials $f_{2},f_{3}\in\mathbb{F}[x,y]$, let $\Gamma_{\mathbb{F}}(f_{2},f_{3})$ denote the bipartite graph with vertex partition $P\cup L$, where $P$ and $L$ are two copies of $\mathbb{F}^{3}$, and $(p_{1},p_{2},p_{3})\in P$ is adjacent to $[l_{1},l_{2},l_{3}]\in L$ if and only if $p_{2}+l_{2}=f_{2}(p_{1},l_{1})$ and $p_{3}+l_{3}=f_{3}(p_{1},l_{1})$. The graph $\Gamma_{3}(\mathbb{F})=\Gamma_{\mathbb{F}}(xy,xy^{2})$ is known to be of girth eight. When $\mathbb{F}=\mathbb{F}_q$ is a finite field of odd size $q$ or $\mathbb{F}=\mathbb{F}_{\infty}$ is an algebraically closed field of characteristic zero, the graph $\Gamma_{3}(\mathbb{F})$ is conjectured to be the unique one with girth at least eight among those $\Gamma_{\mathbb{F}}(f_{2},f_{3})$ up to isomorphism. This conjecture has been confirmed for the case that both $f_{2},f_{3}$ are monomials over $\mathbb{F}_q$, and for the case that at least one of $f_{2},f_{3}$ is a monomial over $\mathbb{F}_{\infty}$. If one of $f_{2},f_{3}\in\mathbb{F}_q[x,y]$ is a monomial, it has also been proved the existence of a positive integer $M$ such that $G=\Gamma_{\mathbb{F}_{q^{M}}}(f_2,f_3)$ is isomorphic to $\Gamma_{3}(\mathbb{F}_{q^{M}})$ provided $G$ has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials $f_2,f_3$ is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials $f_2,f_3$ are characterized completely.