Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

TL;DR

This paper classifies a family of bipartite graphs defined by algebraic equations over the real numbers based on their girth, revealing structural properties related to their algebraic parameters.

Contribution

It provides a complete classification of these algebraically defined monomial graphs according to their girth, a fundamental graph property.

Findings

Complete girth classification of the graphs

Identification of parameter conditions for specific girth values

Insights into the algebraic structure influencing graph girth

Abstract

For positive integers $s,t,u,v$, we define a bipartite graph $\Gamma_{\mathbb{R}}(X^s Y^t,X^u Y^v)$ where each partite set is a copy of $\mathbb{R}^3$, and a vertex $(a_1,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x_1,x_2,x_3]$ in the second partite set if and only if \[ a_2 + x_2 = a_1^s x_1^t \quad \text{and} \quad a_3+x_3=a_1^ux_1^v. \] In this paper, we classify all such graphs according to girth.