Torus Graphs: Theory & Applications

• Torus graphs are a family of structures defined by their embedding on a torus, grid constructions, toric topology labels, or as parametrized graph models.
• They are analyzed using geometric, combinatorial, and spectral methods, with results including parity theorems, morphing algorithms, and rigidity criteria.
• Applications span low-genus embedding theory, periodic discrete models, spatial knotting, optimal routing, and statistical phase-coupling in neural data.

In the literature surveyed here, the expression “torus graphs” is used in several distinct senses rather than as a single standardized term. It may denote graphs embeddable on the torus, torus grid graphs such as $C_m\Box C_n$ and their twisted variants, labeled graphs encoding torus-manifold data, or graph-based families naturally parametrized by a torus, such as magnetic perturbation tori or graphical models on $(S^1)^d$. This diversity reflects three recurrent roles of the torus: as an embedding surface, as a product geometry, and as a parameter space (Biedl, 2022, Qiao, 19 May 2026, Kuroki, 2013, Alon et al., 22 May 2025).

1. Terminological scope

A useful way to organize the subject is to separate the main meanings of the term by the mathematical role played by the torus.

Usage	Defining feature	Representative source
Toroidal graph	A graph admitting a crossing-free embedding on the torus, or a graph of orientable genus $1$	(Biedl, 2022, Afkhami et al., 2014)
Torus grid graph	A periodic grid such as $C_m\Box C_n$ or a twisted torus $TG_m^{(p)}(n)$	(Qiao, 19 May 2026, Đokić et al., 2022)
Torus graph in toric topology	An $n$-valent labeled graph with axial data encoding a torus manifold	(Kuroki, 2013)
Torus-parametrized graph model	A graph-supported family indexed by a torus, such as magnetic perturbations on $T^\beta$ or circular graphical models on $(S^1)^d$	(Alon et al., 22 May 2025, Klein et al., 2019)

This suggests that the phrase is best treated as a family resemblance term. In some areas, the torus is geometric and external to the graph; in others, it is intrinsic to the state space or parameter space attached to the graph.

2. Embeddings on the torus and geometric representation

In low-genus surface theory, a graph is planar if it embeds in the sphere or plane, while in the genus language used for commuting graphs a graph is toroidal when its orientable genus is $1$, meaning that it embeds in the torus $S_1$ but not in the plane (Afkhami et al., 2014). In graph drawing on the flat torus, the phrase toroidal graph is used more broadly for a graph with a crossing-free torus embedding (Biedl, 2022).

The flat torus viewpoint is particularly explicit in the morphing literature. The Euclidean flat torus is modeled as

$(S^1)^d$0

and a geodesic toroidal drawing maps vertices to points of $(S^1)^d$1 and edges to projected line segments. In this setting, geodesics need not be shortest paths. The first algorithm for morphing such drawings shows that any two isotopic essentially 3-connected geodesic toroidal embeddings of the same graph admit a morph through geodesic toroidal embeddings; the morph can be computed in $(S^1)^d$2 time and consists of $(S^1)^d$3 parallel linear morphing steps (Chambers et al., 2020). The proof relies on a toroidal extension of Tutte’s spring embedding theorem and on a geometric analysis of 6-regular toroidal triangulations.

A complementary geometric representation result concerns visibility drawings. Every loopless toroidal graph has a visibility representation on the rectangular flat torus, answering positively a question left open by Mohar and Rosenstiehl and also asked by Tamassia and Tollis (Biedl, 2022). The same paper proves an analogous rectangular-flat representation theorem for graphs on the Klein bottle. Here the torus is not merely a topological host; the rectangle model imposes a rigid axis-aligned geometry, and the proof constructs a cylindrical visibility representation with prescribed path-columns before regluing it into the torus.

At the level of embeddability criteria, the torus admits a parity theorem analogous to the classical Hanani–Tutte theorem for the plane. If a graph can be drawn on the torus so that every pair of independent edges crosses an even number of times, then the graph can be embedded on the torus (Fulek et al., 2020). This establishes the strong Hanani–Tutte theorem for genus $(S^1)^d$4, and it gives a parity characterization of toroidal embeddability that is stronger than existence of a particular embedding.

3. Structural and extremal theories for toroidal graphs

One line of work treats toroidal graphs as exact low-genus objects and asks for classification results. In the setting of commuting graphs of finite groups, the commuting graph $(S^1)^d$5 of a finite non-abelian group $(S^1)^d$6 has vertex set $(S^1)^d$7, with two vertices adjacent exactly when they commute. The toroidal cases are completely classified: $(S^1)^d$8 is toroidal if and only if $(S^1)^d$9 is one of

$1$0

and no finite group has a toroidal non-commuting graph (Afkhami et al., 2014). This classification is obtained by genus obstructions such as $1$1 and $1$2, centralizer structure, and a final finite computation in GAP.

A second structural theory concerns optimal $1$3-toroidal graphs, namely graphs drawn on the torus so that every edge crosses another edge at most once and with exactly $1$4 edges on $1$5 vertices. Their quadrangular subgraph $1$6, consisting of the non-crossing edges, controls both connectivity and matching structure. The possible connectivities are exactly

$1$7

and connectivity $1$8 does not occur. The paper further characterizes the cases $1$9, $C_m\Box C_n$0, and, in the 8-regular case, when $C_m\Box C_n$1 versus $C_m\Box C_n$2. It also proves that an optimal $C_m\Box C_n$3-toroidal graph is $C_m\Box C_n$4-extendable if and only if it is $C_m\Box C_n$5-regular (Koizumi et al., 6 Jan 2025). The torus is essential here because the proofs use annuli, homotopic essential cycles, and toroidal quadrangulations $C_m\Box C_n$6.

Coloring theory provides another exact toroidal classification. Every cyclically 4-edge-connected cubic graph embeddable in the torus is 3-edge-colorable except for the Petersen-like class; equivalently, every nontrivial toroidal snark is obtained from copies of the Petersen graph by dot products, with the first examples being the Petersen graph and one Blanuša snark (Inoue et al., 11 May 2025). The same work derives a strong toroidal nowhere-zero 4-flow theorem: a 2-edge-connected graph embedded in the torus admits a nowhere-zero 4-flow unless it is Petersen-like.

For triangle-free criticality, the torus again yields a sharply constrained structure. There are exactly four irreducible 4-critical triangle-free toroidal graphs, and every 4-critical triangle-free graph drawn in the torus has

$C_m\Box C_n$7

where $C_m\Box C_n$8 is the multiset of face lengths at least $C_m\Box C_n$9; moreover, such a graph has at least seven 4-faces and representativity at least $TG_m^{(p)}(n)$0 (Dvořák et al., 2018). The proof uses reductions by identifying opposite vertices in 4-faces, a finite set of irreducible terminal objects, and computer-assisted enumeration.

4. Toroidal grids and periodic discrete models

A large portion of torus-graph theory concerns discrete periodic grids. The standard torus grid graph is the Cartesian product

$TG_m^{(p)}(n)$1

with vertex set $TG_m^{(p)}(n)$2 and wrap-around adjacency in both coordinates. For the disjunctive domination number $TG_m^{(p)}(n)$3, the general bound for $TG_m^{(p)}(n)$4 is

$TG_m^{(p)}(n)$5

and exact formulas are known for widths $TG_m^{(p)}(n)$6 and $TG_m^{(p)}(n)$7: $TG_m^{(p)}(n)$8

$TG_m^{(p)}(n)$9

These results are obtained by periodic constructions, local counting, and modular obstruction arguments (Qiao, 19 May 2026).

For 2-factor enumeration, the torus grid may be twisted. The graph $n$0 is formed by gluing the first and last columns of $n$1 with a cyclic row shift $n$2, so that $n$3. A common transfer-digraph framework leads to the exact trace formula

$n$4

where $n$5 is the reduced transfer matrix and $n$6 is the rotation matrix on circular binary words (Đokić et al., 2022). The same paper proves the symmetry

$n$7

for the generating functions of 2-factor counts.

Periodic rigidity gives another toroidal grid-like viewpoint. A periodic framework in $n$8 can be modeled as a finite gain graph on the fixed torus

$n$9

with edge gains in $T^\beta$0 recording wrap-around. On a fixed torus, the only trivial infinitesimal motions are translations, so a minimally rigid framework satisfies

$T^\beta$1

rather than the Euclidean Maxwell count $T^\beta$2 (Ross, 2012). The paper also derives subgraph inequalities involving the gain space, showing that torus wrapping contributes directly to rigidity.

The Villarceau torus is a different discrete torus model. It is built on equal-parity vertices in $T^\beta$3, with edges split into acute and obtuse diagonal families. The paper identifies cycles corresponding to toroidal helices, computes poloidal and toroidal revolutions, proves that the graph admits no convex edgecuts and no convex cycles other than 4-cycles, computes the Wiener index, and designs an optimal congestion-balanced routing (Manuel, 2023). This suggests a torus-graph theory oriented toward routing and distance geometry rather than embedding or coloring.

5. Labeled torus graphs and torus-parametrized graph families

In toric topology, torus graphs are a specific labeled combinatorial object. For an omnioriented locally standard $T^\beta$4-dimensional torus manifold, the graph of the orbit space $T^\beta$5 is an $T^\beta$6-valent regular graph $T^\beta$7, and each oriented edge $T^\beta$8 carries an axial function

$T^\beta$9

determined by the omnioriented characteristic function (Kuroki, 2013). The labels satisfy an edge-reversal sign condition, a spanning condition at each vertex, and a compatibility condition via a connection $(S^1)^d$0. In the six-dimensional case $(S^1)^d$1, these are 3-valent labeled graphs, and they classify simply connected 6-dimensional torus manifolds with vanishing odd-degree cohomology up to equivariant diffeomorphism. The classification reduces such manifolds to $(S^1)^d$2, equivariant connected sums of quasitoric manifolds, and $(S^1)^d$3-bundles over $(S^1)^d$4.

A different use of “torus” appears in algebraic geometry of graph hypersurfaces. For a graph $(S^1)^d$5, the projective graph hypersurface

$(S^1)^d$6

is defined by the Kirchhoff polynomial $(S^1)^d$7. If the simplified graph has a fat nexus, then $(S^1)^d$8 admits a nontrivial $(S^1)^d$9-action, with fixed point scheme

$1$0

This implies

$1$1

and over $1$2,

$1$3

for the complement $1$4 (Denham et al., 2020). Here the torus is the algebraic group $1$5, not an embedding surface.

Spectral graph theory yields yet another torus-parametrized family. For a finite connected graph $1$6 with first Betti number

$1$7

magnetic perturbations of a graph-supported Hermitian matrix are parametrized by a $1$8-dimensional torus $1$9. The $S_1$0-th eigenvalue $S_1$1 becomes a function on this torus, and smooth critical points organize into explicit critical submanifolds determined by support and nodal data (Alon et al., 22 May 2025). The Morse index is described by

$S_1$2

where $S_1$3 is the nodal surplus on the support graph. In this sense, the torus is a parameter space attached to the cycle structure of the graph.

6. Spatial, origami, and statistical applications

For spatially embedded torus graphs, knot theory becomes unavoidable. In the A-trail literature motivated by DNA origami, an A-trail is a smooth Eulerian circuit in a graph embedded on a surface. For checkerboard-colorable torus graphs, any A-trail is unknotted, and existence of an A-trail is characterized by associated red and blue covering structures: a smooth transition system is an A-trail exactly when one of the associated covering subgraphs is a covering tree (Morse et al., 2017). This has sharp consequences for torus grids. Aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and such an A-trail is necessarily unknotted; every rectangular torus grid contains an unknotted A-trail, but rectangular grids can also realize any torus knot as an A-trail. The same paper uses gluing to construct infinite higher-genus families with unknotted A-trails and infinite triangular families with none.

Spatial graph theory imposes a complementary restriction. Every nontrivial embedding of a planar graph on the torus contains either a nontrivial knot or a nonsplit link (Barthel, 2014). Equivalently, there are no minimally knotted planar spatial graphs on the torus that avoid both nontrivial knots and nonsplit links whose components are unknots. This excludes toroidal ravels and shows that on the torus, nontriviality of a planar spatial graph must already be visible in a classical knot or link subgraph.

In statistics and neuroscience, torus graphs denote graphical models for multivariate angular data. A random vector of phases

$S_1$4

lives on a $S_1$5-torus, and the torus graph model is the full pairwise exponential family

$S_1$6

with conditional independence characterized by

$S_1$7

(Klein et al., 2019). Because the normalizing constant is intractable, estimation is based on score matching, yielding

$S_1$8

when $S_1$9 is invertible. In the simulations and neural phase-coupling analysis reported there, standard pairwise phase locking value fails to recover conditional structure, whereas torus graphs recover conditional associations and give interpretable cross-region coupling patterns.

Across these settings, torus graphs are unified less by a single formal definition than by a repeated structural theme: the torus introduces periodicity, wrap-around homotopy, and non-Euclidean dependence in ways that force theories different from their planar or Euclidean analogues. That is why torus graphs appear simultaneously in low-genus embedding theory, periodic discrete models, toric topology, spectral theory, knotting, and multivariate phase analysis.