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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 vertexpartition $P\cup L$, 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 andonly 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 tobe 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 ofcharacteristic zero, the graph $\Gamma_{3}(\mathbb{F})$ is conjectured to bethe unique one with girth at least eight among those$\Gamma_{\mathbb{F}}(f_{2},f_{3})$ up to isomorphism. This conjecture has beenconfirmed 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 amonomial 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 theexistence 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 thispaper, these results are shown to be valid when the restriction on thepolynomials $f_2,f_3$ is relaxed further to that one of them is the product oftwo univariate polynomials. Furthermore, all of such polynomials $f_2,f_3$ arecharacterized completely.

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