Coin Graphs, Polyhedra, and Conformal Mapping

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DISCRETE MATHEMATICS

Discrete Mathematics 134 (1994) 133-l 38

Coin graphs, polyhedra, and conformal mapping

Horst Sachs

Institut,fiir Mathematik, Technischr Hochschule Ilmenau, PSF 327, D-6300 Ilmenau, Germaq

Received 18 September 1991; revised 22 May 1992

Abstract

This is a brief report on some interesting theorems and their interconnections. All of these results are known, some of them have only recently been proved. The key result, however, though found already in 1935, was almost forgotten and its original proof remained unnoticed until very recently. Therefore, it seems worthwhile to the author to take the opportunity to inform the graph theoretic community about some facts which appear to be not so well known.

1. Definitions

A coin is a closed circular disk in the (euclidean) plane, a coin system is a finite set of coins (not necessarily of equal radii) with disjoint interiors (i.e., a coin system is a finite circle packing). The contact scheme C of a coin system S is a graph G = C(S) whose vertices and edges are in one-to-one correspondence with the coins of S, and with those pairs of coins of S which are tangent, respectively. Graph G is called a coin graph iff G is the contact scheme of some coin system; clearly, any coin graph is finite, simple, and planar. If S is a coin system and G a graph isomorphic to C(S) then we shall say that G is realized by S. By a stereographic projection mapping the plane onto the unit sphere (and conversely), any coin system is transformed into what we will call a spherical coin system, i.e., a finite spherical circle packing (and conversely). Graph G is called polyhedral iff it is isomorphic to the l-skeleton of a convex (three-dimensional) polyhedron. Recall that graph G is polyhedral if and only if G is j&e, simple, planar, and 3-connected [23]. An egg is a strictly convex body in E3 with a smooth boundary.

2. Four remarkable theorems

Theorem 1. Every jnite, simple, planar graph is a coin graph.

Theorem 2. Let G and G be a pair of dual polyhedral graphs. G and G can simulta- neously be realized by coin systems S and S, respectively, such that S and Shave the same points of contact and the boundary circles of the coins of S and those of the coins of S intersect perpendicularly in the common contact points. The simultaneous realization of G and c is unique up to circle preserving transformations of the plane (which depend on 6 real parameters).

Theorem 3. Given a convex polyhedron P, there is a convex polyhedron Q combinatorially equivalent to P such that all edges of Q are tangent to the unit sphere.

Clearly, the tangency cones of the unit sphere S whose vertices are the vertices of Q touch S in circles which determine a spherical coin system, Z, say, and the faces of Q intersect S in circles which also determine a spherical coin system, cQ say; both these systems have the same contact points (namely, the points in which S is touched by the edges of Q) in which the boundary circles of their (spherical) coins intersect perpendicularly. Note that each of these coin systems determines Q. In this sense, Q is unique up to circle preserving transformations of the unit sphere (which depend on 6 real parameters).

Theorem 4. Given an egg E and a convex polyhedron P, there is a convex polyhedron Q , combinatorially equivalent to P, all of whose edges are tangent to E. Iffor some integer k 3 1 the boundary of E is Ck” -smooth and has positive Gaussian curvature everywhere, then the space of such polyhedra Q is a six-dimensional Ck-smooth manifold.

Clearly, Theorem 4 implies Theorem 3. Let Q be a convex polyhedron whose edges are tangent to S. Replace every edge e of Q by a straight line e’ which is perpendicular to e and touches S at the same point as e: then the new lines determine (the l-skeleton of) a convex polyhedron Q’ whose edges are tangent to S; evidently, Q and Q’ are duals of each other, thus Q’s Q and cB = Co. Applying a stereographic projection, we immediately see that Theorem 2 and The- orem 3 are equivalent. The simultaneous realization of G and G by orthogonal coin systems, and the corresponding representation of P (and its dual) as described above, being unique up to a group r of circle preserving transformations, Theorem 3 provides a polyhedron representation which is canonic module r. As for the proof of Theorem 1, it is not difficult to see that it suffices to prove Theorem 1 for 3-connected triangulations. Thus Theorem 1 follows from Theorem 2:

Theorem 4-+Theorem 30Theorem 2+Theorem 1.

3. Comment

Theorem 1 was formulated as a conjecture in 1978 by Wegner [26] and in 1984 by Jackson and Ringel [ll, 161. Theorem 3 was conjectured also in 1984 by H. Sachs/ Discrete Mathematics 134 (1994) 133-138 135

Griinbaum and Shephard; this conjecture is contained in an interesting paper of Schulte [22] on the general inscribability problem and also - among other interesting problems, conjectures and propositions on convex polyhedra - in a paper of Griinbaum and Shephard (191. In July 1990, the author (who at that time had no knowledge of [22] or [9]) reformulated Theorem 3 as an open problem [14]. Within the framework of a general topological theory of 3-manifolds, a proof of Theorem 1 (that can be extended to a proof of Theorem 2) was given by Thurston (7251; this proof which is essentially based on results of Andreev [1,2] (1970) was already mentioned in Thurston’s talk given at the International Congress of Mathematicians (ICM), Helsinki, 1978. Convergency problems connected with this proof were investigated by Colin de Verdi&e [5]. In July 1990, the author found a (relatively) simple approach to Theorem 1 using methods of conformal mapping: Theorem 1 may be considered as resulting from a certain limiting case of Paul Koebe’s ‘Kreisscheibentheorem’ (‘Kreisnormierun- gsprinzip’) [12] (1908) (see also [S, Ch. VI). However, discussing his ideas with Rainer Kiihnau (University of Halle) the author learned that his result is not new: in fact, Theorem 1 was already discovered in 1935 by Koebe himself [13] who found this theorem as a corollary to a general theorem on conformal mapping of ‘contact domains’. These are Koebe’s words [13, p. 1621:

“Die Aufgabe, auf der Kugeloberjltiche n Kreisjltichen, deren GroJe unbekannt bleibt, nebeneinander ohne gegenseitige iiberdeckung so zu lagern, daJ sie ein durch ein beliebiges gewiihnliches Triangulationsschema vorgeschriebenes Kontaktschema er- fiihen (SchlieJungsproblem), gestattet immer eine und, abgesehen von einer Kreisver- wandtschaft, nur eine L&sung.”

Koebe added the following footnote:

“Auf diesen SchlieJIungssatz bzw. einen damit zusammenhiingenden merkwiirdigen Polyedersatz beabsichtige ich in einer besonderen Note zuriickzukommen, die ich der Preuj. Akademie der Wissenschaften iiberreichen will.”

This clearly means that Koebe anticipated Theorem 3, too. However, the paper whose publication he announced in the above quoted footnote has never appeared - probably owing to his serious illness and the circumstances of World War II. Koebe died in 1945. In 1990/91 there has been much progress in the theory of coin graphs most of which came to the author’s attention only after he had finished the first version of his manuscript. Colin de Verdiere [6,7] generalized Theorem 1 in various directions (including infinite coin systems) combining a variational principle with the ‘method of coherent angles’. A paper containing elementary algorithmic proofs of Theorems 1 and 2 has been prepared by Pulleyblank and Rote [ 151; in this paper the authors also describe a remarkable application of Theorem 1 to a class of combinatorial optimization problems, including the travelling salesman problem, for planar graphs. Another simple proof of Theorem 2 was obtained by Brightwell and Scheinerman [4]; for some 136 H. Sachs: Discrete Mathematics 134 11994) 133-138

related results, see also [20]. Also, Gil Kalai from Jerusalem was recently reported to have found a proof of Theorem 3. Theorem 4 - the most general of the results discussed here - was proved by Schramm [21]; the tools used are taken from differential topology. It should also be mentioned that in a very recent paper He and Schramm [lo] have given another far-reaching generalization of Theorem 1:

“Let .F be a triangulation of ajnite genus open surface with at most countably many ends. Then there is a closed Riemann surface [w and a circle packing P c K! whose graph is combinatorially equivalent to the l-skeleton of F and whose carrier is a circle domain in R. Moreover, [w and P are unique up to conformal or anti-conformal homeomorphisms.” ([lo, Theorem 1.41)

It is most remarkable that not only Theorem 1 can be proved using methods from the theory of conformal mapping but that, conversely, Theorem 1 also provides a means for ‘discretizing’ conformal mapping, thus allowing new approaches to the central theorems of this theory (Schwarz’s Lemma, Riemann’s Mapping Theorem, Koebe Uniformization, etc.: see [3,10,17-19,241; more references can be found in [lo]). Thus Theorem 1 - which originally seemed to be of a more or less special character, an isolated theorem having aesthetic appeal rather than promising useful- ness or applicability - has, in a sense, proved to be of central significance and, together with its generalizations, has given rise to a renaissance of certain parts of the theory of conformal mapping (where already important new results have been found, see, e.g., [lo]). What makes this topic so fascinating to the author is the fact that in it combinatorial and differential topology, discrete geometry, hyperbolic geometry, the theory of polyhedra, graph theory, analysis (conformal mapping) and even combinatorial optimization are found intrinsically interwoven.

Note added in proof. During the last two years, the theory of circle packings (coin graphs) and its generalizations and applications to conformal mapping and other fields have developed rapidly. The following brief account is by no means complete. New proofs of Theorems 1,2 and 3 have been given by Darmet (Grenoble) [7a] and Carter and Rodin [4a]. Various interesting generalizations and applications of the ‘circle packing theorem’ (i.e., Theorem 1) are due to Schramm [21a-21d]. Complexity problems connected with circle packings have been discussed by Mohar [14a]. Not to forget a cumulative bibliography compiled in 1993 by Stephenson [24a].

Acknowledgements

The author thanks Yves Colin de Verdi&e, A. Darmet, Branko Griinbaum, Francois Jaeger, Heinz Adolf Jung, Rainer Kiihnau, Jentj Lehel, Bojan Mohar, Bill Pulleyblank, Gerhard Ringel, Giinter Rote, Edward R. Scheinerman, Wolfgang SchGne, Oded Schramm and Gerd Wegner for fruitful discussions and valuable information.

References

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