A discrete GaussBonnet type theorem
Abstract.
We discuss a curvature theorem for subgraphs of the flat triangular tessellations of the plane. These graphs play the analogue of ”domains” in two dimensional Euclidean space. We show that the Pusieux curvature satisfies , where is the Euler characteristic of the graph, is the boundary of and where the arc length of the sphere of radius in . This formula can be seen as a discrete GaussBonnet formula or Hopf Umlaufsatz.
Key words and phrases:
Graph theory, GaussBonnet, Curvature1991 Mathematics Subject Classification:
Primary: 05C10 , 57M15Dedicated to Ernst Specker to his 90th birthday.
1. Introduction
For a domain in the plane with smooth boundary , the GaussBonnet formula or Umlaufsatz
relates the curvature of the boundary curve
with the Euler characteristic of
the region. For a simply connected region for which the boundary is a simple closed
curve, the total boundary curvature is . This GaussBonnet type result is a form of Hopf’s Umlaufsatz
and relates a differential geometric quantity, the boundary curvature,
with a topological invariant, the Euler characteristic.
In differential geometry, curvature needs a differentiable structure, while Euler characteristic does not.
It is the transcending property between different mathematical branches which makes GaussBonnet
type results interesting.
We prove here a discrete version of a ”Hopf Umlaufsatz” [3] which is of combinatorial nature;
curvature is an integer. The result applies to special two dimensional graphs which are part of a flat two dimensional
background graph , where the
dimensionality is defined inductively. While the Euler characteristic is a topological notion,
we need ”smoothness assumptions” to equate the total boundary curvature with the Euler characteristic.
The curvature, we consider here is , where is the arc length of the sphere
at the point . The sphere is a subgraph of with vertices of all points of distance and edges
consisting of pairs in such that and have distance .
As we will explore elsewhere, for many compact twodimensional graphs without boundary, like triangularizations
of polyhedra with 5 or 6 faces, the integral of the curvature
over the entire graph is . The ”smoothness” assumptions are more subtle than
corresponding results for . For the later, the result is
essentially a reformulation of Euler’s formula and holds for any ”two dimensional graph” with or without boundary.
We will look at a relation between the ”first order curvature” and second order curvature at the end of
this article.
The main result in this paper is the formula
which holds for discrete domains and for a second order curvature .
To do so, we need to specify precisely what a ”smooth domain” is.
The background lattice plays here the role of the twodimensional plane. Its vertices can be realized as
the set of points . The edges are formed by the set of
pairs for which the Euclidean distance is . In the infinite graph , every point has 6 neighbors. Together with edges formed
by neighboring vertices, these points form the unit sphere , a subgraph of .
Similarly, any sphere of radius in this discrete plane has length .
The curvature is zero at every point of the background lattice .
Figure: Curvature computation. The numbers near each vertex indicate the curvature of the point. At each of a few chosen points, we have drawn the spheres of radius and in . Adding up the curvatures over the boundary gives 12. If a point has as a neighborhood a disc of radius , the curvature is zero.
2. Topology of the planar triangular lattice
A finite subset of the triangular lattice defines a graph , where is the set of vertices in and where is a subset of edges in , pairs in have distance within . We start by defining a dimension for graphs. To our best knowledge, this notion seems not yet have appeared, even so in the graph theory literature, several notions of dimension exist. The definition of dimension is inductive and rather general and does not require the graph to be a subset of .
Definition 1.
A sphere in the subgraph of whose vertices consists the set of points in which have geodesic distance to , normalized, so that adjacent points have distance within . The edges of the sphere graph are all pairs with for which is in . A disc in the graph is the set of points which have distance in .
Definition 2.
A vertex of a graph is called 0dimensional, if is not connected to any other vertex. A subset of is called dimensional if every point of is dimensional in . Zerodimensionality for a graph means that it has no edges. A point of is called 1dimensional if is dimensional, where is the unit sphere of within . A finite subset of is called 1dimensional if any of the points in is dimensional. A point of is called dimensional, if is a onedimensional graph. A subset of is called dimensional, if every vertex of is a dimensional point.
The dimension does not need to be defined. For example, a point which has a sphere which contains of one and zero dimensional
components has no dimension. One could define inductively a fractional dimension
by adding to the average fractional dimensions of the points on the unit sphere.
As an illustration of the notion of dimension, lets look at the platonic solids as graphs. The cube and the dodecahedron are one dimensional. The isocahedron and octahedron are two dimensional. The tetrahedron is three dimensional, because the unit sphere of each point is 2 dimensional. The cube and dodecahedron become two dimensional after kising (stellating) their faces. The tetrahedron becomes 2 dimensional after truncating corners.
Definition 3.
A point in called an interior point of if the sphere in the graph is the same than the sphere in the background lattice . In other words, for an interior point, the sphere is a onedimensional graph without boundary.
Definition 4.
A point of a twodimensional graph is a boundary point of , if it is not an interior point in but has a neighbor in which is an interior point.
For an interior point, the sphere is a closed circle, for a boundary point, the sphere is a union of onedimensional arcs.
Definition 5.
The boundary of is the set of boundary points of . The interior of is the set of interior points of .
Remarks:
a) The set of subsets defines a
topology on such that the interior of is open and the boundary is closed.
b) The interior of a two dimensional graph is not necessarily a 2 dimensional graph.
The disc of radius in for example is has a single interior point so that the interior is zerodimensional.
c) Two dimensionality of a graph has no relation with ”being planar”.
There are planar graphs like the tetrahedral graph which is three dimensional in our sense but which is planar.
And there are graphs like triangularizations of a torus, which are two dimensional but not planar.
d) Topologically, one can show that the triangular graph is the only simplyconnected twodimensional
flat graph without boundary.
Definition 6.
We call a subset of a domain if the following 5 conditions are satisfied:
The conditions (i),(ii), (iii) are natural. Condition (iv) assures that no unnatural ”fissures” can exist.
Condition (v) assures that the connectivity topology of the domain and the connectivity topology
of the interior set are the same.
Definition 7.
A domain is called a finite domain, if it is a finite graph which is a domain. A domain is called a smooth domain, if it is a domain and its complement is a domain too.
Remarks:
a) We could additionally require the interior of a domain to be twodimensional but we do not need that.
Actually, the proof of the main theorem becomes simpler if we do not make this assumption. It would just
lift a difficulty on a different level. For us it will be important to look at the dimension of points
in the interior of .
b) Some of these conditions for ”domains” have analogues in the continuum, where they are necessary
for the classical GaussBonnet to be true:
we can not have ”hairs” sticking out of the domain for example.
The closure of the complement of a domain is a domain
too and we can not just leave out part of the boundary. Also in the continuum, it should not happen that
parts of domains are tangent to each other. We also can not allow the boundary to be twodimensional,
like for the Mandelbrot set.
c) For a smooth domain, we can look at the interior of the complement of .
Then, the boundaries satisfy . The three sets ,
and partition the graph .
Figure: Examples of domains. The number in the upper right corner is the total boundary curvature of the domain.
Figure: Examples of graphs which are not domains. To the left, a set with 2, 1 and 0 dimensional points. It violates conditions (i). The second example is a set with both 2 and 1 dimensional points.
Figure: The left example is a twodimensional set with no interior points and no boundary points. It violates condition (ii). The second one violates (v).
Figure: Examples of graphs which are not domains. The first violates (v), the third violates (iii).
Figure: Domains which are not smooth domains. The curvature of the first was computed
while assuming the nearest neighbor connection to be an edge as required by condition (iv).
If the connection is not in place (violating (iv)), the total curvature would be 24.
The following lemma allows us to deal more efficiently with eligible regions and eliminates many subsets which are not regions. It says that the set of interior points determines the region as well as its boundary.
Lemma 1.
Let be a domain and be the set of interior points of . Then , where is the disc of radius in . Especially, the interior set determines the domain completely.
Proof.
If a point is in , then it is either an interior point or a point adjacent to an interior point. Therefore . On the other hand, if is in , then for some . Because and by definition of being an interior point, we have . ∎
Remark: For a simply connected region, also the boundary of determines the region, but we do not need that.
3. Curvature
Definition 8.
Let denotes the number of edges in the sphere . We call it the arc length of the sphere .
Remarks.
a) Note that is not necessarily the number of vertices in .
Similarly, is the number of edges in which is not always equal to
the number of vertices in .
b) The sphere does not necessarily have to be connected, nor does it have
to have a defined dimension. It could be a union of a segment and a point for example.
Definition 9.
The curvature of a boundary vertex in a region is defined as
The curvature of a finite domain is the sum of the curvatures over the boundary.
Remarks.
a) This definition is motivated by differential geometry since one can derive an
analogue formulas in the continuum
for a point on the
boundary curvature of a region.
b) Note that as defined, refers to the
geodesic circle of radius in and not in so that every point of distance
in to belongs to whether there is a connection within from to or not.
The reason for this choice is that we do want the curvature definition to be nonlocal.
This subtlety will not matter since for the definition of smooth curve, we anyhow disallow
situations where points have a large distance within but small distance in .
Figure: The first picture is a smooth domain. It is not simply connected although the two parts of the regions are at first separated enough to get a curvature 24. In the second case ”feels” part of the other region and the curvature is not a multiple of .
Figure: In the first picture, the domain is still not smooth because the complement is not a domain. The last example is a smooth domain. It has become simply connected.
Definition 10.
A curve in a smooth domain is a sequence of points in the interior of such that and consequently is an edge of . A curve is a closed curve if . In graph theory, a curve is called a chain. It is a nontrivial closed curve if its length is larger than . It is called a simple closed curve, if all points are different and .
Definition 11.
A domain is called simply connected if every closed curve in the interior of can be deformed to trivial closed curve within , where a deformation of a curve within consists of a composition of finitely many elementary deformation steps with and such that are in . As in the continuum, simply connectedness means that any closed curve in the interior of can be deformed to a point within the interior of .
4. The curvature 12 theorem
Our main result of this paper is a discrete version of the ”Umlaufsatz”. It will be generalized to more general domains below.
Theorem 2 (Curvature 12 Umlaufsatz).
The total boundary curvature of a finite, smooth and simply connected domain is .
Proof.
For the proof, it suffices to look at local deformations. We start with an arbitrary simply connected smooth region and find a procedure to remove interior points near the boundary while keeping the simply connectedness property and keeping also the curvature the same. Removing one point only affects the curvatures in a disc of radius so that only finitely many cases need to be studied:
Lemma 3 (Curvature is local).
Let be two regions and be a point in both and . Let and be the discs of radius in and respectively. Define . If , then
In other words, if we remove a point from a region, then the total curvaturechange can be read off from
the curvaturechanges in a disc of radius .
We could check all possible configurations in discs of radius and compare the total curvature
before and after the center point is removed. We indeed checked with the
help of a computer that in all cases, where the total boundary curvature changes, the number of local
connectivity components of the interior has changed or the complement has become nonsmooth near the
removed point. These experiments helped us also to get the conditions what a domain is.
But checking all possible local deformations is not a proof. We also need to know that there is always a point
which we can remove without changing the topology of or its complement.
It turns out that this question is of more global nature. Take a ring shaped region for example which has a one dimensional interior. No point can be removed without the curvature to change. The key is to look at the dimension of points in the interior of and distinguish points which are onedimensional in and points which are twodimensional in . A zero dimensional interior means for a simply connected region that the graph is the disc of radius in . By removing interior points, we want to reach this situation.
Figure: Pruning a tree, a simply connected domain. To reduce a region, we have to trim the tree,
removing alternatively twodimensional interior points and onedimensional interior points until only
one interior point is left.
We are allowed to look at the topology of interior points because defines by the above
lemma, it is enough to check what happens if we remove interior points. Our goal is to show:
Proposition 4 (Trimming a tree).
For any simply connected smooth region for which the interior set has more than one point, it is possible to remove an interior point from , such that the new region defined by remains a simply connected smooth region with one interior point less and such that the curvature does not change.
The theorem follows from this proposition. Lets introduce some terminology:
Definition 12.
Given a smooth, simply connected region with interior . Denote by the points in which are one dimensional in . Similarly, call the set of points in which are two dimensional in . Connected components of are called either branches or bridges. Connected components of are called ridges. A branch of is a connected component of for which at least one point has only one interior neighbor. All other connected components of are called bridges.
Figure: A simply connected region with ridges, bridges. All branches have been pruned. Now, we have to start etching the ridges. We have the choice of 4 end ridges here. The simply connectivity assures that there is an end ridge.
Figure: The only situation, where we can
not trim any more one dimensional branches nor twodimension ridges.
The set is the union of points which are twodimensional in and points which are
onedimensional in . We will use two procedures called pruning and
etching to make the region smaller. The pruning procedure removes a onedimensional interior point at branches.
The etching procedure removes a twodimensional interior point at ridges.
Figure: Pruning reduces the lengths of branches. Since curvature is local, we only need to
check for a few end situations that the total curvature does not change.
Lets start with the pruning procedure which removing interior points which are onedimensional in . It allows us to remove onedimensional branches until we can no more reduce onedimensional points in . Removing onedimensional parts will make sure that there will be a twodimensional ridges ready for the etching procedure. Here are the situations which can occur locally at a point of a branch.
Figure: A onedimensional point which has 1 interior neighbor. After removing a boundary point, we end up with region 0.
Figure: Reducing a one dimensional point at the boundary. For any of the two situations, we end up with a region with one interior neighbor.
Figure: For any of the
first 3 situations, we end up with two neighboring interior points.
After reducing one dimensional branches, the tree still can have one dimensional parts: these are
2D ridges connected with onedimensional bridges which can not be pruned without changing the topology.
Figure: Etching thins out ridges. The etching is done at ridges which are end ridges, where only
one bridge is attached. With too many bridges attached, the etching process might not work.
The etching procedure
is invoked if no onedimensional branches are left. The region consists now of
twodimensional ridges connected with bridges.
Our goal is to see that we can remove a twodimensional interior point of a ridge.
The simply connectivity
implies that there is a ridge which has only one bridge connected to it. To see this, look at a new
graph, which contains the twodimensional ridges as vertices and onedimensional bridges as edges.
This graph has no closed loops and is connected and must be a tree with at least one end points. We can consequently
focus our discussion to such an endridge for which only one dimensional bridge is attached.
We are able to remove a boundary point on the opposite side of that region, where no branches can be and
where the boundary is ”smooth”.
Figure: Reducing a twodimensional interior point at the boundary
which has 2 interior points as neighbors.
Figure: Reducing a twodimensional point at the boundary
which has 3 interior points as neighbors.
Figure: A situation where the point has 3 interior neighbors
and where the point can not be reduced.
Figure: A situation, where the point has 4 interior neighbors and where
the point can not be removed.
Figure: A situation, where the point has 4 neighbors which are interior points and where
the point can not be removed.
Figure: Four interior points bounding an interior point. The middle point
can not be removed while keeping the region a smooth region.
Figure: Having exactly 5 interior points bounding an interior point
is not possible. We then necessarily have 6 neighbors.
Figure: A bridge and branches. For the picture with the bridge, no interior point which is
onedimensional in can be removed.
For the picture with the branches, no interior point which is
twodimensional in can be removed. This is a situation, where the branches
first need to be trimmed.
Once the etching process is over, we can again start pruning branches, or we are
left with a region with only one interior point.
If a region can no more be pruned and edged then consists of only one point and
consists of only points and in this case, we know the total curvature is .
Since pruning and etching did not change the curvature and we have demonstrated that one can reduce down every simply connected region to a situation with one interior point, this completes the proof of the curvature 12 theorem. ∎
5. Discrete GaussBonnet
To generalize the Umlaufsatz to domains which are not necessarily simply connected we first define the Euler characteristic of a region using Euler’s formula:
Definition 13.
A face in a domain is a triangle of 3 points in for which all three points have mutual distance . An edge in is a pair of points in of distance . A vertex is a point in . Denote by the number of faces in , by the number of edges and the number of vertices. The Euler characteristic of the domain is defined as .
Example: for a simply connected region, the Euler characteristic is .
Lemma 5.
The Euler characteristic does not change under the pruning and etching operations defined above: both removing an end point of a one dimensional branch, as well as removing a two dimensional point from a ridge does not change it.
Proof.
The number of interior points of a smooth region is and which can be proved by adding faces: each face added is equivalent to adding 2 edges. ∎
Figure: A region, where no interior point can be removed any more and which has more than
one interior point is not simply connected.
Remark. The Euler characteristic of and is the same if is a smooth region.
Theorem 6 (Discrete GaussBonnet theorem).
If is a finite smooth domain with boundary , then
We could use the same pruningetching technique as before. However, pruning and etching can lead
to final situations which have no end points like a ring. Instead of classifying
all these final situations, it is easier
to reduce the general situation to a simply connected situation.
There are two ways, how to change the topology:

build bridges between different connected components.

fill holes to make the region simply connected
Merging different unconnected components is no problem. As long as their complement is a smooth
region too, both the Euler characteristic as well as the total curvature add up.
1. We can assume the region to be connected, because both curvature as well as Euler characteristic are additive with respect to adding disjoint domains. To illustrate this more, we can also join two separated regions along with a one dimensional bridge. The curvature drops by , the number of connected components drops by .
Figure: Joining two regions changes the total curvature by 12.
Definition 14.
The interior of a hole is a bounded simply connected smooth region such that is a component of the complement of . By definition a whole and the region share a common part of the boundary.
By the Umlaufsatz for simply connected regions, the hole has total curvature 12. A key observation is that the pointwise curvatures at the inner boundary of enclosing the hole are just the negative of the corresponding pointwise curvatures of the hole. This follows almost from the definition of curvature and the fact that the circles and so that .
Figure: Filling a simply connected hole from a larger region adds to the curvature exactly the
same amount than the total curvature of the hole. The reason is that the pointwise curvatures of the removed
inside region matches exactly the curvatures of the inner outside region
if the inside region and the outside region have a common onedimensional boundary.
This shows that if we fill a hole, the total curvature increases by . Simultaneously, the
Euler characteristic increases by .
Alternatively, we could also cut rings:
Figure: When cutting a ring, the curvature changes from 0 to 12. Only the
last region is a smooth region. The second last is a region but not smooth
because the complement is not a region.
6. Compact flat graphs
If we introduce identifications in the hexagonal background graph , the topology of the background space changes. Identifying points along two parallel lines for example produces a flat cylinder. With a triangular tiling, we can tessellate a torus. Because there is no boundary now, the sum of the curvatures is zero, which is the Euler characteristic. Note that there are many different nonisometric graphs which lead to such tori. We call them twisted tori. As graphs they are different even if the number of faces, edges and vertices are fixed.
Figure: A flat torus obtained by identifying opposite sides of a rectangular
domain in a hex lattice. The total curvature is zero.
The notion of regular domain can be carried over to discrete manifolds like the twisted tori just mentioned. Let be such a twisted background torus. We assume that it is large enough so that is a circle at every point. Let be a subgraph of defined as before. We still have:
Theorem 7.
If is a domain in a background torus , then
Remark: More flat compact graphs can be obtained using ”worm hole” constructions. Let be a possibly twisted torus as defined above and let , be two points for which the discs and are disjoint and the spheres and are circles. Any orientable graph isomorphism between and produces an identification of points in . Without identification, the total curvature of the boundary of the domain is because every removed disc produces curvature .
7. Combinatorial curvature
In this section, we consider a more elementary GaussBonnet formula. The curvature is again defined by a Puisaux discretization, but only circles of length appear in the definition. It is a first order curvature.
Definition 15.
For a two dimensional graph with boundary, we define the combinatorial Puiseux curvature as
for interior points and
for boundary points.
For this combinatorial curvature, GaussBonnet is much easier. For subgraphs
of hexagonal lattice
the boundary curvature is almost trivially equal to because the curvature is related to angles
of the corresponding polygon: for a boundary point, is the interior angle of the polygon.
Because by the polygonal version of the Umlaufsatz, we have .
Actually, GaussBonnet holds in great generally for arbitrary twodimensional graphs with or without boundary.
Since it is so closely related to the Euler characteristic, we should the attribute it to Euler,
even so we are not aware that Euler considered , nor that he looked at the dimension of a graph.
Theorem 8 (Combinatorial GaussBonnet).
Assume is a twodimensional finite graph for which the boundary is either empty or forms itself a one dimensional set. Then
Remarks.
1) This result does not need the rigid requirements on the ”domain” as before nor does the graph have to be part of ;
it works for any twodimensional graph, with or without boundary.
2) The result appears in a different formulation, which does not make its GaussBonnet nature evident:
the Princeton Companion to Mathematics [2] mentions on page 832 the formula
, where is the number hedral faces and summation is
over all faces. This is an equivalent formulation, but it makes the GaussBonnet character
less evident.
Note that the combinatorial GaussBonnet theorem is entirely graph theoretical. It avoids the pitfalls with the definition, what a polyhedron is [4]. (Common definitions of ”polyhedra” refer to an ambient Euclidean space or impose additional structure on a graph). We can take a general finite graph which is twodimensional at each point. Its points are either boundary points, points where the unit sphere is onedimensional but not closed, or an interior points, where the unit sphere is a circle, a simple closed graph without boundary.
Proof.
Assume first that the graph has no boundary. For a twodimensional graph, all faces necessarily are triangles. Therefore, the number of faces and the number of edges are related by the dimensionality formula
Furthermore, we have the edge formula
which is obtained by counting edges in a different way.
Using the definition of the Euler characteristic, and these two formulas, we compute
This finishes the proof in the case of a graph without boundary.
The case with boundary can be reduced to the boundaryless case: The boundary is a union of closed cycles. For each of these cycles just add an other point and add edge connections from each of the cycle boundary points of to . This produces a graph without boundary and which contains as a subgraph. The formula without boundary shows that is the sum of curvatures of the original interior points and the sum of the curvatures of the boundary points as well as the sum of the curvatures to each newly added point :
We also have
For boundary points, and so that . From the previous boundary less case, we get
so that
∎
The just verified combinatorial GaussBonnet is entirely graph theoretical. Our curvature definition was motivated from the notion of Jacobi fields in the classical case given by second derivatives. While ”smoothness” requirements” are necessary for the more sophisticated GaussBonnet formula, the just mentioned metric GaussBonnet holds for any polyhedron with triangular faces. For example, every finite triangularization of a twodimensional compact manifold works. Let us explain a bit more, why the curvature
is ”differential geometric”:
the GaussJacobi equations in differential
geometry require the second differences of a Jacobi field .
[1].
Our starting point had been to extend Jacobi fields to the discrete for numerical purposes:
for a discretized Jacobi field with smallest space step , we have . The Jacobi equations suggest to call this . Since is the variation of the
geodesic when changing the angle, we can integrate over the circle and we get the length of the circle
of radius . Therefore is a multiple of and there is no reason to normalize this in the discrete.
The ”first order curvature” on the other hand only requires first order differences.
The curvature has some advantages over the curvature :

the curvature formula for the boundary and in the interior is the same, while for the curvature , one has to distinguish boundary and interior.

there is no reference to a flat background structure for , while refers to the flat situation with via integers or .

it can be generalized to more general situations, where the distance in the graph can vary and where we have no natural flatness as a reference. We can look for example for distance functions which minimize the total curvature.

it is more closely rooted to differential geometry of manifolds and classical notions like Jacobi fields, a notion which is of ”second order” too.

it can be adapted to higher dimension, when defining scalar curvature for graphs and where no natural ”flat triangulated ambient reference graph” exists.
To summarize, we think that while is ”metric”, has a more ”differential geometric” flavor. Similarly as
many metric results extend to the differential geometric setup, things are more restricted also in the
discrete, if higher order difference notions are used. The limitations of the results are related to similar
limitations we know in the continuum.
We can combine the two results: for the Puiseux curvature with radius defined by
we get the following corollary:
Corollary 9 (K2 formula).
If is a twodimensional smooth domain in the triangular tessellation of the plane, then
Proof.
Since and , we get by addition . The left hand side is the combinatorial Puiseux curvature for radius . ∎
Note that unlike the combinatorial curvature formula , the formula is only obvious modulo the main result for ”smooth domains” proved here. If we wanted to establish GaussBonnet type results for curvatures like , the restrictions on discrete domains would be even more severe.
References
 [1] M. Berger. A Panoramic View of Riemannian Geometry. Springer Verlag, Berlin, 2003.
 [2] T. Gowers (Editor). The Princeton Companion to Mathematics. Princeton University Press, 2008.
 [3] H. Hopf. Über die Drehung der Tangenten und Sehnen ebener Curven. Compositio Math., 2, pages 50–62, 1935.
 [4] I. Lakatos. Proofs and Refutations. Cambridge University Press, 1976.