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arXiv · 2512.21963

Topological properties of generalized Markoff mod $p$ graphs

Shohei Satake·Yoshinori Yamasaki
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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.

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Shohei Satake, Yoshinori Yamasaki. 2025-12-26. Topological properties of generalized Markoff mod $p$ graphs. https://arxiv.org/abs/2512.21963

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