Map Color Theorem Research Articles

A simple proof of the Map Color Theorem for nonorientable surfaces

We give a more simple proof of the Map Color Theorem for nonorientable surfaces that uses only four constructions of current graphs instead of 12 constructions used in the previous proof. For every i=0,1,2,3, using an index one current graph with cyclic current group, we construct a nonorientable triangular embedding of K12s+3i+1 that can be easily modified into a nonorientable triangular embedding of K12s+3i+3 and a minimal nonorientable embedding of K12s+3i+5. As a result, for every n≥10, n∉{11,14,20}, we construct a minimal nonorientable embedding of Kn. During the modifications, all but 2(4s+i) faces of the nonorientable triangular embedding of K12s+3i+1 become faces of the nonorientable triangular embedding of K12s+3i+3, and all but 2(4s+i)+1 faces of the nonorientable triangular embedding of K12s+3i+3 become faces of the minimal nonorientable embedding of K12s+3i+5.

Simultaneous current graph constructions for minimum triangulations and complete graph embeddings

The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify previous constructions by Ringel, Youngs, Guy, and Jungerman. All these new constructions are index 3 current graphs sharing nearly all of the structure of the simple solution for Case 5 of the Map Color Theorem.

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $$\chi _s$$ of simplicial complexes naturally generalize the chromatic number $$\chi _1$$ of a graph. In any fixed dimension d, the s-chromatic number $$\chi _s$$ of d-complexes can become arbitrarily large for $$s\le \lceil d/2\rceil $$ (Bing in The geometric topology of 3-manifolds, Colloquium Publications, vol 40, American Mathematical Society, Providence, 1983; Heise et al. in Discrete Comput Geom 52:663–679, 2014). In contrast, $$\chi _{d+1}=1$$ , and only little is known on $$\chi _s$$ for $$\lceil d/2\rceil <s\le d$$ . A particular class of d-complexes are triangulations of d-manifolds. As a consequence of the Map Color Theorem for surfaces (Ringel in Map color theorem, Grundlehren der mathematischen Wissenschaften, vol 209, Springer, Berlin, 1974), the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $$\chi _2$$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $$\chi _2$$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $$f=(127,8001,5334)$$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction (Heise et al. 2014) along with embedding results (Bing 1983) can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of Heise et al. (2014), we obtain a rather small triangulation of the 3-dimensional sphere $$S^3$$ with face vector $$f=(167,1579,2824,1412)$$ and 2-chromatic number 5.

Quadrangular embeddings of complete graphs and the Even Map Color Theorem

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph Kn for n≡5(mod8), and nonorientable ones for n≥9 and n≡1(mod4). These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph Kn, n≥4, the minimum genus, both orientable and nonorientable, for the surface in which Kn has an embedding with all faces of degree at least 4, and also for the surface in which Kn has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph G has a perfect matching and a cycle then the lexicographic product G[K4] has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.

K-Chromatic Number of Graphs on Surfaces

A well-known result (Heawood [Quart. J. Pure Appl. Math., 24 (1890), pp. 332–338], Ringel [Map Color Theorem, Springer-Verlag, New York, 1974], Ringel and Youngs [Proc. Nat. Acad. Sci., U.S.A., 60 (1968), pp. 438–445]) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Euler genus of S. A partition of a graph G into k parts consists of k edge-disjoint subgraphs $G_1,\dots,G_k$ such that $E(G)=E(G_1)\cup E(G_2)\cup\dots\cup E(G_k)$. The k-chromatic number $\chi_k (G)$ is the maximum of $\sum_{i=1}^k\chi(G_i)$ over all partitions of G into k parts. We derive a Heawood-type formula for the k-chromatic number of graphs embedded in a fixed surface, improving the previously known upper bounds. In infinitely many cases, the new upper bound coincides with the lower bound obtained from embedding disjoint cliques in the surface. In the proof of this result, we derive a variant of Euler's formula for the union of several graphs that might be interesting independently.

Reformulating the map color theorem

This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou–Kryuchkov conjecture, the Penrose formula, the vector cross-product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown.

A map colour theorem for the union of graphs

In 1890 Heawood [Map colour theorem, Quart. J. Pure Appl. Math. 24 (1890) 332–338] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g ⩾ 1 . This upper bound became known as the Heawood number H ( g ) . Almost a century later, Ringel [Map Color Theorem, Springer, New York, 1974] and Ringel and Youngs [Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA 60 (1968) 438–445] proved that the Heawood number H ( g ) is in fact the maximum chromatic number as well as the maximum clique number of graphs embedded on a surface of Euler genus g ⩾ 1 besides the Klein bottle. In this paper, we present a Heawood-type formula for the edge disjoint union of two graphs that are embedded on a given surface Σ . More precisely, we determine the number H 2 ( Σ ) such that if a graph G embedded on Σ is the edge disjoint union of two graphs G 1 and G 2 , then ω ( G 1 ) + ω ( G 2 ) ⩽ χ ( G 1 ) + χ ( G 2 ) ⩽ H 2 ( Σ ) . Similar to the results of Ringel and Ringel and Youngs, we show that this bound is sharp for all but at most one non-orientable surface Σ .

Another Proof of the Map Color Theorem for Nonorientable Surfaces

The simplest known proof of the Map Color Theorem for nonorientable surfaces (obtained by Youngs, Ringel et al. and given in Ringel's book “Map Color Theorem”) uses index one and three current graphs, and index two and three inductive constructions. We give another proof, still using current graphs, but simpler than Youngs' and Ringel's in several ways. Our proof uses only index one current graphs and no inductive constructions. For every h⩾8, h≠9, 14, we construct an index one current graph Γ( h) that yields a minimal nonorientable embedding ψ( h) of K h . The current graphs Γ( h) have the property that Γ( n) and Γ( n+1) are not too different from each other and share common ladder-like fragments. As a result, the embeddings ψ( n) and ψ( n+1) have a large number of common faces: it is shown that, as n approaches infinity, for n≢3, 9 mod 12 (resp. n≡3,9 mod 12) no less than 5/8 (resp. 5/16) of all faces of ψ( n) appear in ψ( n+1).

Dirac's map-color theorem for choosability

It is proved that the choice number of every graph G embedded on a surface of Euler genus e ≥ 1 and e ≠ 3 is at most the Heawood number $H(\epsilon)= \lfloor(7+\sqrt{24\epsilon+1})/2\rfloor$ and that the equality holds if and only if G contains the complete graph KH(e) as a subgraph. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 327–339, 1999

Dirac's map-color theorem for choosability

Dirac's map-color theorem for choosability

On a generalized family of colorings

Motivated by a question in cellular telecommunication technology, we investigate a family of graph coloring problems where several colors can be assigned to each vertex and no two colors are the same within any ball of radiusR. We find bounds and coloring algorithms for different kinds of graphs including trees,n-cycles, hypercubes and lattices. We briefly examine connections to Heawood's map color theorem and state a few conjectures and open problems.

Heawood's empire problem

Heawood's map color theorem is generalized to maps of empires on surfaces.

Tight and 0-tight polyhedral embeddings of surfaces

In [1] T. Banchoff has studied the problem of high condimensional tight polyhedral embeddings of closed surfaces into Euclidean space. He gave an upper bound for the essential codimension depending on the Euler characteristic of the surface. In [1] and [5] he proved that this bound is attained in some cases and that it is not attained for the Klein bottle. In the present paper we show that this bound is sharp in any case (except the Klein bottle) and that for each surface there exist tight substantial embeddings into Euclidean space of arbitrary dimension up to the Banchoff upper bound. The proof depends essentially on the Heawood map color theorem proved by G. Ringel and J.W.T. Youngs. In addition we get similar results for tight and 0-tight embeddings of surfaces with boundary where it may be remarkable that in case of tight surfaces with one boundary component the Banchoff upper bound can be improved.

Index four orientable embeddings and case zero of the Heawood conjecture

By imposing a special symmetry, we are able to construct index four triangular embeddings of graphs in compact orientable 2-manifolds. Because of the complexity of the current graphs required, such embeddings have heretofore been unattainable, but the imposed symmetry reduces the problem to constructing a special kind of index two current graph. We illustrate the method with a solution for case zero of the Heawood conjecture, using an abelian group, thus completing a constructive proof of the Heawood map color theorem, and eliminating the need for Galois field theory and nonabelian groups in its solution. The method has also been used in the determination of the genus of K n, n, n, n .

The combinatorial map color theorem

AbstractThis paper is written in the spirit of the author's book: Map Color Theorem (1974). We try to develop the Map Color Theorem in a combinatorial way, circumventing the unwieldy embedding theory. Similar (but not identical) generalizations have recently and independently been developed by Alpert (in press) and by Stahl (in press). The first nine theorems are one‐dimensional versions of known facts from the theory of two‐dimensional compact manifolds. Theorems 10 to 13 are to my knowledge completely new results.

A new solution for the nonorientable case 1 of the heawood Map Color Theorem

A simpler method for proving γ(K n) = (n − 3)(n − 4) 6 when n ≡ 1 (mod 12).

Book Review: Map color theorem

Book Review: Map color theorem