Topological properties of generalized Markoff mod $p$ graphs

TL;DR

The paper studies the topological properties of generalized Markoff mod $p$ graphs $ mathcal{G}_{\kappa}(p)$ defined by $x^2+y^2+z^2=xyz+\kappa$ over $\mathbb{F}_p$, focusing on non-planarity, embeddability, and short cycles. It develops a framework to construct $K_{3,3}$-subdivisions via sextuples of Markoff triples, employing Chebyshev polynomials, resultants, Gröbner bases, and the Chebotarev density theorem to obtain explicit subdivisions and density results for primes $p$. The main contributions include explicit $K_{3,3}$-subdivisions for infinitely many primes with natural densities (at least $1/2$ in general and at least $13/16$ for most $\kappa$), mutual disjoint subdivisions within the giant component, and genus- and embedding-related results, notably that ${\mathcal{G}(p)}$ is toroidal/projective-planar only for $p=7$. These results support the view that the giant component $\mathcal{C}_{\kappa}(p)$ exhibits expander-like global structure and have implications for related cryptographic hash constructions built from Markoff-type graphs.

Abstract

The generalized Markoff mod $p$ graph is defined via the equation $x^2+y^2+z^2=xyz+κ$ 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.

Figures (11)

• Figure 1: A sextuple of Markoff triples
• Figure 2: Illustrations of $K_{(\alpha_{\kappa},-1)}\in \mathcal{K}^{\text{prop}}_{1,\kappa}(p)$ for $(\kappa,p)=(6,13)$ (left), and $(\kappa,p)=(8,13)$ (right). The $X_i$-$Y_j$-paths (of length $2$) and the $X_i$-$Y_i$ paths in $K_{(\alpha_{\kappa},-1)}$ are depicted using red and blue edges, respectively, for $1\le i,j\le 3$.
• Figure 3: Illustrations of $K_{(\alpha,\beta)}\in \mathcal{K}^{\text{prop}}_{2,\kappa}(p)$ for $(\kappa,p)=(8,19)$ with $(\alpha,\beta)=(12,14)$ (left), and $(\kappa,p)=(-5,19)$ with $(\alpha,\beta)=(10,14)$ (right). The $X_i$-$Y_j$-paths (of length $4$) and the $X_i$-$Y_i$ paths in $K_{(\alpha,\beta)}$ are depicted using red and blue edges, respectively, for $1\le i,j\le 3$.
• Figure 4: Illustrations of $K_{(\alpha,\beta)}\in\mathcal{K}^{\text{prop}}_{3,\kappa}(p)$ for $(\kappa,p)=(5,13)$ with $(\alpha,\beta)=(0,10)$ (left), and $(\kappa,p)=(7,13)$ with $(\alpha,\beta)=(0,7)$ (right). The $X_i$-$Y_j$-paths (of length $6$) and the $X_i$-$Y_i$ paths in $K_{(\alpha,\beta)}$ are depicted using red and blue edges, respectively, for $1\le i,j\le 3$.
• Figure 5: Illustrations of $K_{(\alpha,2)}\in\mathcal{K}^{\mathrm{prop}}_{(p-1)/2,\kappa}(p)$ for $(\kappa,p)=(7,11)$ (left), and $(\kappa,p)=(8,11)$ (right) with $\alpha=2+\sqrt{\kappa-4}$. The $X_i$-$Y_j$-paths (of length $p-1$) and the $X_i$-$Y_i$ paths in $K_{(\alpha,2)}$ are depicted using red and blue edges, respectively, for $1\le i,j\le 3$.

Theorems & Definitions (53)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: Theorem \ref{['thm-disjoint-k']}
• Theorem 1.4: Theorem \ref{['thm-proj']}
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 3.1