[Paper Review] On enumeration of a class of toroidal graphs

Dipendu Maity, A. Upadhyay|arXiv (Cornell University)|Nov 1, 2013

Finite Group Theory Research3 references3 citations

TL;DR

This paper presents a complete classification of semi-equivelar maps of eight specific types on the torus—{3³,4²}, {3²,4,3,4}, {3,6,3,6}, {3⁴,6}, {4,8²}, {3,12²}, {4,6,12}, and {3,4,6,4}—using a systematic algorithmic approach based on T(r,s,k) representations and cycle type analysis. The authors enumerate non-isomorphic maps up to 24–60 vertices, establishing isomorphism criteria via cycle lengths and providing explicit tables of classified maps for small vertex counts.

ABSTRACT

We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We know the classification of the maps of types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$ on the torus. In this article, we attempt to classify maps of types $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$ on the torus.

Motivation & Objective

• To classify all non-isomorphic semi-equivelar maps of eight specific types on the torus.
• To develop an algorithmic framework for enumerating such maps based on graph representations and cycle structure.
• To determine isomorphism classes by analyzing lengths of non-contractible cycles of specific types in T(r,s,k) models.
• To provide explicit tables of non-isomorphic maps for small vertex counts (up to 24–60 vertices) across all types.
• To complete the classification of all 11 types of semi-equivelar maps on the torus, resolving a long-standing open problem in topological graph theory.

Proposed method

• Represent maps using a T(r,s,k) parameterization, where r, s, k define grid-like embeddings on the torus.
• Define and classify cycles of specific types (e.g., A₁, A₂, Z₁) that correspond to homology classes and structural features.
• Use cycle length analysis to establish isomorphism invariants: two maps are isomorphic iff their cycle length pairs match under permutation.
• Derive admissible ranges for r, s, k based on topological and combinatorial constraints (e.g., divisibility, vertex count, cycle existence).
• Apply lemmas to prove existence and uniqueness conditions for each map type under given r, s, k constraints.
• Generate and tabulate non-isomorphic maps by checking cycle length pairs and parameter validity across vertex counts.

Experimental results

Research questions

• RQ1 How can semi-equivelar maps of types {3³,4²}, {3²,4,3,4}, {3,6,3,6}, {3⁴,6}, {4,8²}, {3,12²}, {4,6,12}, and {3,4,6,4} be systematically enumerated on the torus?
• RQ2 What are the necessary and sufficient conditions on parameters r, s, k in the T(r,s,k) representation for such maps to exist?
• RQ3 How can isomorphism between two such maps be determined algorithmically using cycle structure?
• RQ4 What is the complete set of non-isomorphic maps of each type for small vertex counts (≤24–60 vertices)?
• RQ5 Can the classification of all 11 types of semi-equivelar maps on the torus be completed through this method?

Key findings

• The paper completes the classification of all 11 types of semi-equivelar maps on the torus, with explicit results for eight types.
• For type {4,8²}, maps exist only when r is divisible by 4, s ≥ 1, and rs ≥ 20, with k constrained modulo r based on s.
• Maps of type {4,8²} with 20 vertices yield two non-isomorphic classes: T(20,1,6) and T(20,1,14), both with cycle lengths (20,20).
• For 24 vertices, four non-isomorphic maps exist: two from T(24,1,k) and two from T(8,3,k), with cycle length pairs (24,8) and (24,24).
• The classification is fully algorithmic: two maps are isomorphic iff their non-homologous cycle length pairs match under permutation.
• Tables 1–8 list all non-isomorphic maps for each type up to 22–60 vertices, with explicit T(r,s,k) representations and cycle data.