On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

Abstract

For a field $\mathbb{F}$ and functions $f,g,h,j\colon\mathbb{F}\to \mathbb{F}$, we define $\Gamma_\mathbb{F}(f(X)h(Y),g(X) j(Y))$ to be a bipartite graph where each partite set is a copy of $\mathbb{F}^3$, and a vertex $(a,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x,x_2,x_3]$ in the second partite set if and only if \[a_2+x_2=f(a)h(x) \quad \text{and} \quad a_3+x_3=g(a)j(x).\] In this paper, we completely classify all such graphs by girth in the case $h=j$ (subject to some mild restrictions on $h$). We also present a partial classification when $h\neq j$ and provide some applications.