Topological properties of generalized Markoff mod $p$ graphs

TL;DR

This paper studies the topological features of generalized Markoff mod p graphs, focusing on properties like non-planarity and cycle structures, using a combination of graph theory, algebra, and number theory techniques.

Contribution

It introduces a systematic method for analyzing topological properties of these graphs, including the construction of $K_{3,3}$-subdivisions, advancing understanding of their structure.

Findings

Demonstrates non-planarity of the graphs.

Identifies conditions for surface embeddability.

Finds existence of short cycles in the graphs.

Abstract

The generalized Markoff mod $p$ graph is defined via the equation $x^2+y^2+z^2=xyz+\kappa$ over the finite field $\mathbb{F}_p$ of prime order $p$. In this paper, we investigate the topological properties of the graph such as non-planarity, surface embeddability, and the existence of short cycles. Our approach is based on a systematic construction of $K_{3,3}$-subdivisions, integrating techniques from graph theory, computer algebra, and number theory.