## Topological Graph Theory -- A Survey (1996)

Venue: | CONG. NUM |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Archdeacon96topologicalgraph,

author = {Dan Archdeacon},

title = {Topological Graph Theory -- A Survey},

journal = {CONG. NUM},

year = {1996},

volume = {115},

pages = {115--5}

}

### OpenURL

### Abstract

### Citations

1258 |
Graph Theory with Applications
- Bondy, Murty
- 1976
(Show Context)
Citation Context ...ial In this section we introduce some of the basic terms and concepts of topological graph theory. The reader seeking additional graph-theoretic definitions should consult the book by Bondy and Murty =-=[44]-=-. A more detailed treatment of embeddings is in the book by Gross and Tucker [103]. We examine in turn the basic terms, surfaces, Euler's formula and its consequences, the maximum and minimum genus, c... |

376 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context .... It would be interesting to find either a common extension of or relationship between Heawood and Grotzsch's theorems. No good characterization is to be expected, since Garey, Johnson and Stockmeyer =-=[93]-=- showed that in general the problem is NP-complete. We refer the reader to Steinberg [222, 223] for a recent survey of this three color problem. The 2-color problem is easy. A plane graph is 2-colorab... |

364 |
Graphs minors II. Algorithmic aspects of tree-width
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...the width of the graph. The n-grid has large tree-width [191]. Hence any graph which contains the n-grid as a minor also has large tree-width. The Tree-Width Theorem asserts the converse. Theorem 7.3 =-=[190]-=- There exists a function f such that a graph G has tw(G)sf(k) if and only if it has a k-grid minor. As a partial result toward's Wagner's minor conjecture, Robertson and Seymour [192] were able to sho... |

307 |
Topological Graph Theory
- Gross, Tucker
- 1987
(Show Context)
Citation Context ...cal graph theory. The reader seeking additional graph-theoretic definitions should consult the book by Bondy and Murty [44]. A more detailed treatment of embeddings is in the book by Gross and Tucker =-=[103]-=-. We examine in turn the basic terms, surfaces, Euler's formula and its consequences, the maximum and minimum genus, combinatorial descriptions of embeddings, and partial orders. 2.1 Basic Terms A gra... |

292 |
Graph Coloring Problems
- Jensen, Toft
(Show Context)
Citation Context ...ctured that for simple graphs it is bounded above by \Delta + 2. This conjecture is known to be true for several classes of graphs. Various general upper bounds are also known. We refer the reader to =-=[123]-=- for a discussion of the known results. The conjecture is true for plane graphs except for the cases \Delta = 6; 7. The low degree cases are due to Rosenfeld [202] (\Delta = 3) and Kostochka [136, 137... |

275 |
Every planar map is four-colorable
- Appel, Haken
- 1976
(Show Context)
Citation Context ...ing that every cubic graph was Hamiltonian. Petersen [172] clarified the relation with edge-colorings and introduced his famous graph (see [113]). The Four-Color Theorem was proved by Appel and Haken =-=[3, 4]-=- in 1977. The proof was at first controversial, in part because of the reliance on long computer calculations. However, the result has been proven several times independently, most recently by Roberts... |

252 |
Graph minors XIII. The disjoint paths problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ...ructure of the graph. For example Arnborg [21] has found polynomial time algorithms for k-coloring and Hamiltonicity of graphs of bounded tree width. A fundamental result due to Robertson and Seymour =-=[195]-=- is the following solution to the k-path problem. 40 Theorem 7.4 For each fixed k there exists a polynomial time algorithm for deciding if a graph G with vertices x 1 ; y 1 ; : : : ; x k ; y k has k d... |

165 |
Sur le problème des courbes gauches en topologie, Fund
- Kuratowski
(Show Context)
Citation Context ...hich embed on a fixed surface. 4.1 Characterizing Planar Graphs An important early question is to characterize those graphs which embed on the plane. One such characterization was given by Kuratowski =-=[145]-=- in 1930. The same theorem was proven independently and roughly concurrently by Frink and Smith [87], who never published their paper after hearing of Kuratowski's proof. This theorem is very importan... |

152 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, Proskurowski
- 1989
(Show Context)
Citation Context ... number of problems which are NP-complete in general are polynomial for graphs of bounded tree-width. Here the algorithm is able to exploit the "tree-like" structure of the graph. For exampl=-=e Arnborg [21]-=- has found polynomial time algorithms for k-coloring and Hamiltonicity of graphs of bounded tree width. A fundamental result due to Robertson and Seymour [195] is the following solution to the k-path ... |

136 |
Towards an architectureindependent analysis of parallel algorithms
- Papadimitriou, Yannakakis
- 1990
(Show Context)
Citation Context ...f the book and that edges lie entirely in one page. These restrictions are useful in the applications to VLSI layouts, where the pages can represent circuit boards, or queues used in scheduling tasks =-=[171]-=-. Define the page-number of a graph pn(G) as the minimum number of pages need to draw G in this manner. Since two pages form R 2 , one might guess that every planar graph has page-number 2. However, i... |

119 |
the tree theorem, and Vazsonyi's conjecture
- Well-quasi-ordering
- 1960
(Show Context)
Citation Context ... implications of the Robertson-Seymour Theorem are very important. We examine these two aspects in the following subsections. 38 Figure 4: A graph of tree-width three 7.1 Trees and Tree-Width Kruskal =-=[143]-=- proved in 1960 that rooted finite trees were well-quasi-ordered under the topological containment order. It follows that there are also no infinite antichains in this collection under the minor order... |

95 |
On acyclic colorings of planar graphs
- Borodin
- 1979
(Show Context)
Citation Context ...s \Delta = 6; 7. The low degree cases are due to Rosenfeld [202] (\Delta = 3) and Kostochka [136, 137, 138] (\Delta = 4; 5). These cases do not use planarity. The high degree cases are due to Borodin =-=[56]-=- (\Deltas9) and Andersen [2] (\Delta = 8) and do use planarity. An entire coloring of a plane graph colors the vertices, faces, and edges simultaneously so that adjacent or incident elements receive d... |

91 |
A Theorem on Planar Graphs
- Tutte
- 1956
(Show Context)
Citation Context ...ned a C b2w=3c \Theta C b2w=3c minor. We mention the following on cycles in graphs unrelated to homotopy. Whitney proved that every 4-connected plane triangulation is Hamiltonian. A number of authors =-=[17, 29, 83, 229, 240, 243, 244, 228]-=- have investigated generalizations of this concept, relaxing the connectivity, replacing planarity with locally planar on other surfaces, and replacing Hamiltonian with various types of walks. We refe... |

88 |
Map Color Theorem
- Ringel
- 1974
(Show Context)
Citation Context ...and the orientablility or nonorientability of the surface, was completed by Ringel and Youngs [186] in 1968 (see esp. [265] for the nonorientable case). A nice account of the proof is given in Ringel =-=[181]-=-. Conversely, it is hard in the plane to prove that four colors suffice for all maps. However, on other surfaces (except the projective plane) it is easy to show that the conjectured number of colors ... |

85 |
Graph minors. III. Planar tree-width
- Robertson, Seymour
- 1984
(Show Context)
Citation Context ...n the grid. A subdivision of the graph is now a minor of the grid after contracting all edges within the vertex disks. The n-grid is related to the width of the graph. The n-grid has large tree-width =-=[191]-=-. Hence any graph which contains the n-grid as a minor also has large tree-width. The Tree-Width Theorem asserts the converse. Theorem 7.3 [190] There exists a function f such that a graph G has tw(G)... |

84 |
The book thickness of a graph
- Bernhart, Kainen
- 1979
(Show Context)
Citation Context ...uch harder question and is still unknown. The best known bound [164] is pn(K n;m )sd(2n +m)=4e. We refer the reader to the seminal works by Chung, Leighton, and Rosenberg [72] and Bernhart and Kainen =-=[38]-=- for further discussions on the subject. 28 5.4 Relative Embeddings As noted in Section 2.5 any embedding can be represented in terms of a rotation scheme and a signature. In this section we examine e... |

81 |
Every planar graph is 5-choosable
- Thomassen
- 1994
(Show Context)
Citation Context ...sts of size k to the vertices has a list coloring. The list chromatic number of the plane is the maximumsl (G) over all planar G. Theorem 3.1 The list chromatic number of the plane is five. Thomassen =-=[236]-=- shows sufficiency with an elegant proof that every planar graph is list 5-colorable. Voigt [251] gives an example of a planar graph and a list assignment of four colors which can not be list colored.... |

74 |
Über graphen und ihre anwendung auf determinantentheorie und mengenlehre
- König
- 1916
(Show Context)
Citation Context ...reader to Steinberg [222, 223] for a recent survey of this three color problem. The 2-color problem is easy. A plane graph is 2-colorable if and only if every face is bounded by a walk of even length =-=[133, 134]-=-. The 1-color problem is not quite pointless, but it is edgeless. Coloring the faces of a plane graph is equivalent to coloring the vertices of the dual. What if we try to color the vertices and faces... |

68 |
Non-separable and planar graphs
- Whitney
- 1932
(Show Context)
Citation Context ...e orthogonal. A graph G is an algebraic dual of a graph G if there is a function OE : E(G) ! E(G ) such that C is a cycle of G if and only if OE(C) is a cocycle of G . The following is due to Whitney =-=[258, 259]-=-. Theorem 4.3 A graph is planar if and only if it has an algebraic dual. In fact, if G and G are algebraic duals, then there exists an plane embedding of G so that G is the geometric dual. Let G be a ... |

65 |
Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel
- Grötzsch
- 1959
(Show Context)
Citation Context ...r graphs. When do three? We mention two famous 3-color theorems. In 1898 Heawood [111] proved that a plane triangulation is vertex 3-colorable if and only if every vertex was of even degree. Grotzsch =-=[104]-=- proved that every triangle-free planar graph was vertex 3colorable. (My favorite proof of this theorem is due to Thomassen [237].) Thomassen [241] gives a list version of Grotzsch's Theorem. Kr'ol [1... |

65 |
How Colorful the Signed Graph
- Zaslavsky
- 1984
(Show Context)
Citation Context ... of a signed graph can be interpreted as preordaining the orientation preserving orientation reversing walks. For the general theory of signed embeddings we refer the reader to the works of Zaslavsky =-=[268, 269, 270]-=-. We mention three particular results of interest to the author. Zaslavsky [266] has determined the maximum Euler genus among all signed graphs on n vertices where every edge is signed negatively. Thi... |

59 |
Knots and links in spatial graphs
- Gordon
- 1983
(Show Context)
Citation Context ...e bounds a disk disjoint from the rest of the graph. Note that flat implies linkless and knotless. 30 When does a graph have a linkless, knotless, or flat embedding? Sachs [204] and Conway and Gordon =-=[73]-=- showed the following. Theorem 5.3 Any embedding of K 6 in 3-space contains a pair of disjoint cycles which are linked. Robertson, Seymour, and Thomas [197, 198] proved that a graph admits a flat embe... |

57 |
The graph genus problem is NP-complete
- Thomassen
- 1989
(Show Context)
Citation Context ...aphs [67]. We close our discussion on the minimum and maximum genus of a graph with the computational aspects. Theorem 4.8 Determining the minimum orientable genus of a graph is NPcomplete (Thomassen =-=[230]-=-). There is a polynomial-time algorithm to find the maximum orientable genus of a graph (Furst, Gross and McGeoch [88]). The techniques of [230] extend easily to show that determining the minimum nono... |

49 |
A Kuratowski theorem for the projective plane
- Archdeacon
- 1981
(Show Context)
Citation Context ...S. Graphs in I(S) are called irreducible since they do not embed in the surface but any proper subgraph (or minor) does embed. The first result for a surface other than the plane is due to Archdeacon =-=[5, 6]-=-. Theorem 4.9 There are exactly 103 graphs topological irreducible graphs for the projective plane. The graphs were originally found by Glover, Huneke, and Wang [77] (Neil Robertson found the 103 rd g... |

48 | Embedding graphs in books: a layout problem with applications to VLSI design
- Chung, Leighton, et al.
- 1987
(Show Context)
Citation Context ...ite graph is surprisingly a much harder question and is still unknown. The best known bound [164] is pn(K n;m )sd(2n +m)=4e. We refer the reader to the seminal works by Chung, Leighton, and Rosenberg =-=[72]-=- and Bernhart and Kainen [38] for further discussions on the subject. 28 5.4 Relative Embeddings As noted in Section 2.5 any embedding can be represented in terms of a rotation scheme and a signature.... |

44 |
Graph minors IV. Tree-width and well-quasiordering
- Robertson, Seymour
- 1990
(Show Context)
Citation Context ...rse. Theorem 7.3 [190] There exists a function f such that a graph G has tw(G)sf(k) if and only if it has a k-grid minor. As a partial result toward's Wagner's minor conjecture, Robertson and Seymour =-=[192]-=- were able to show that the class of graphs of bounded treewidth were well-quasi-ordered under minors. Using that result and a structure theorem they were able to prove the general case [196]. 7.2 Alg... |

43 |
Graph minors. VII. Disjoint paths on a surface
- Robertson, Seymour
- 1988
(Show Context)
Citation Context ...rameter measures the local planarity of the embedding. Observe that the face-width of an embedding is equal to the face-width of the dual. The face-width was first introduced by Robertson and Seymour =-=[193]-=-. The first extensive study of this parameter was by Robertson and Vitray [200]. Several special cases have been commonly used. For example, an embedding of a connected G is of fws1 if and only if it ... |

43 |
Graphs, Groups and Surfaces
- White
- 1973
(Show Context)
Citation Context ...[256]; for a particularly nice proof see Stahl and White [221]. The orientable genus of regular quadripartite graphs is due to Garmen [94] and Jungerman [125] (the special case K 4(3) is due to White =-=[255]-=-). The orientable genus of the cube has been found by several authors [185, 33, 100]. The nonorientable genus is due to Jungerman [126]. For low dimensional cubes we note that the formula for the orie... |

40 |
103 graphs that are irreducible for the projective plane
- Glover, Huneke, et al.
- 1979
(Show Context)
Citation Context ...n the plane is due to Archdeacon [5, 6]. Theorem 4.9 There are exactly 103 graphs topological irreducible graphs for the projective plane. The graphs were originally found by Glover, Huneke, and Wang =-=[77]-=- (Neil Robertson found the 103 rd graph). Archdeacon proved that the list was complete. Vollmerhaus [252] independently verified this completeness, unaware of Archdeacon's work. These 103 topological ... |

40 |
A theorem on paths in planar graphs
- Thomassen
- 1983
(Show Context)
Citation Context ...ned a C b2w=3c \Theta C b2w=3c minor. We mention the following on cycles in graphs unrelated to homotopy. Whitney proved that every 4-connected plane triangulation is Hamiltonian. A number of authors =-=[17, 29, 83, 229, 240, 243, 244, 228]-=- have investigated generalizations of this concept, relaxing the connectivity, replacing planarity with locally planar on other surfaces, and replacing Hamiltonian with various types of walks. We refe... |

40 |
Embeddings of graphs with no short noncontractible cycles
- Thomassen
- 1990
(Show Context)
Citation Context ...y null in the surface. Such a walk is necessarily a simple cycle. In fact, it is the shortest simple cycle which does not bound a disk in the surface. This parameter was first introduced by Thomassen =-=[231]. The-=- dual-width, dw(G), is the edge-width of the dual embedding. The face-width fw(G) is the minimum n = C"G taken over all noncontractible C in the surface. A cycle C achieving this minimum can be c... |

39 |
P.: Graph minors. VIII. A Kuratowski Theorem for General Surfaces
- Robertson, Seymour
- 1990
(Show Context)
Citation Context ...by Wagner that any infinite set of graphs contains one which is a minor of another. This much stronger conjecture implies the Erdos-Konig conjecture since no two irreducible graphs are comparable. In =-=[194]-=- they gave a proof of this for graphs of bounded genus, which implies both the orientable and nonorientable cases. They have since given a proof of Wagner's conjecture in general [196]. Their proofs a... |

37 |
Solution of the Heawood map-coloring problem
- Ringel, Youngs
- 1968
(Show Context)
Citation Context ...th the exception of K 7 in Klein's bottle). That task, broken into 24 cases by the residue of n modulo 12 and the orientablility or nonorientability of the surface, was completed by Ringel and Youngs =-=[186]-=- in 1968 (see esp. [265] for the nonorientable case). A nice account of the proof is given in Ringel [181]. Conversely, it is hard in the plane to prove that four colors suffice for all maps. However,... |

35 |
The Petersen Graph
- Holton, Sheehan
- 1993
(Show Context)
Citation Context ...ght he had solved the 4-color problem; his mistake was believing that every cubic graph was Hamiltonian. Petersen [172] clarified the relation with edge-colorings and introduced his famous graph (see =-=[113]-=-). The Four-Color Theorem was proved by Appel and Haken [3, 4] in 1977. The proof was at first controversial, in part because of the reliance on long computer calculations. However, the result has bee... |

32 |
Graphs and their chromatic numbers
- Behzad
- 1965
(Show Context)
Citation Context ...eceive distinct colors. The total chromatic number 00 (G) is the minimum number of colors needed. It is bounded below by \Delta + 1 where \Delta is the maximum degree. Independently Behzad and Vizing =-=[39, 248]-=- conjectured that for simple graphs it is bounded above by \Delta + 2. This conjecture is known to be true for several classes of graphs. Various general upper bounds are also known. We refer the read... |

32 |
Polyhedral decompositions of cubic graphs
- Szekeres
- 1973
(Show Context)
Citation Context ...(the same techniques lead to a second such graph of the same order). In 1948 Blanche Descartes found [78] an example on 210 vertices. It was not until 1973 that a fourth example was found by Szekeres =-=[226]-=-. At this time several powerful construction techniques were developed by Issacs [118], yielding two infinite classes of snarks. One of these classes, the flower snarks, had been 12 discovered indepen... |

31 |
On the geographical problem of the four colors
- Kempe
(Show Context)
Citation Context ...of the four-color-conjecture is that a number of the most important contributions to the subject were originally made with the belief that they were solutions." One of the first of these was by K=-=empe [130]-=- who introduced the recoloring methods now known as Kempe chains. Heawood [110] pointed out the error in Kempe's argument, but was able to modify it to give a correct proof that every planar graph was... |

31 |
Embedding planar graphs in four pages
- Yannakakis
- 1989
(Show Context)
Citation Context ... along the spine so that the given graph was a spanning subgraph of a Hamiltonian graph. This cannot always be done (although it is true for triangle-free graphs). The best bound is due to Yannakakis =-=[263]-=- who showed that any planar graph can be embedded in a book with 4 pages, and that 4 pages were sometimes necessary. The page-number of K n is dn=2e. The lower bound can be easily seen since in any or... |

29 |
Genus distributions for bouquets of circles
- Gross, Robbins, et al.
- 1989
(Show Context)
Citation Context ...edding distribution was first introduced by Gross and Furst [101]. The complete embedding distribution is known only for a few small graphs and for a few infinite classes. The latter include bouquets =-=[102]-=- (see also [177, 217]), closed ended ladders [89], and cobblestone paths [89]. 41 It is interesting to note that all known embedding distributions are unimodal, in fact, strongly unimodal. It is conje... |

29 |
Grötzsch’s 3-color theorem and its counterparts for the torus and the projective plane
- Thomassen
(Show Context)
Citation Context ...x 3-colorable if and only if every vertex was of even degree. Grotzsch [104] proved that every triangle-free planar graph was vertex 3colorable. (My favorite proof of this theorem is due to Thomassen =-=[237]-=-.) Thomassen [241] gives a list version of Grotzsch's Theorem. Kr'ol [139, 140] noted that a plane graph is 3-colorable if and only if it is a subgraph of an Eulerian triangulation. It would be intere... |

28 |
Trees in Polyhedral Graphs
- Barnette
- 1966
(Show Context)
Citation Context ...ned a C b2w=3c \Theta C b2w=3c minor. We mention the following on cycles in graphs unrelated to homotopy. Whitney proved that every 4-connected plane triangulation is Hamiltonian. A number of authors =-=[17, 29, 83, 229, 240, 243, 244, 228]-=- have investigated generalizations of this concept, relaxing the connectivity, replacing planarity with locally planar on other surfaces, and replacing Hamiltonian with various types of walks. We refe... |

27 |
The decline and fall of Zarankiewicz’s Theorem
- Guy
- 1968
(Show Context)
Citation Context ... more difficult is the problem of demonstrating lower bounds, of stating that every drawing must have a particular number of crossings. The bound on cr(K n ) is known to be exact for ns12 (see, e.g., =-=[106]-=-). The bound on cr(K n;m ) is exact for ns6 [132] and for n = 7, ms10 [260]. The bound on cr(C n \Theta Cm ) is exact for n = 3 [109, 175, 178], for n = 4 [82, 35], and for n = 5 [175, 224]. In each c... |

26 |
Hierarchy for imbedding-distribution invariants of a graph
- Gross, Furst
- 1987
(Show Context)
Citation Context ...om (here the embeddings are all orientable). One goal is to study the distribution over all embeddings of the genus of the surface. This embedding distribution was first introduced by Gross and Furst =-=[101]-=-. The complete embedding distribution is known only for a few small graphs and for a few infinite classes. The latter include bouquets [102] (see also [177, 217]), closed ended ladders [89], and cobbl... |

25 |
Cyclic-order graphs and Zarankiewicz’s crossing-number conjecture
- Woodall
- 1993
(Show Context)
Citation Context ...at every drawing must have a particular number of crossings. The bound on cr(K n ) is known to be exact for ns12 (see, e.g., [106]). The bound on cr(K n;m ) is exact for ns6 [132] and for n = 7, ms10 =-=[260]-=-. The bound on cr(C n \Theta Cm ) is exact for n = 3 [109, 175, 178], for n = 4 [82, 35], and for n = 5 [175, 224]. In each case the first reference (or two) gives the proof for m = n, while the last ... |

24 |
Genus distributions for two classes of graphs
- Furst, Gross, et al.
- 1989
(Show Context)
Citation Context ...and Furst [101]. The complete embedding distribution is known only for a few small graphs and for a few infinite classes. The latter include bouquets [102] (see also [177, 217]), closed ended ladders =-=[89]-=-, and cobblestone paths [89]. 41 It is interesting to note that all known embedding distributions are unimodal, in fact, strongly unimodal. It is conjectured that this is always the case [102]. Short ... |

24 |
Intersections of curve systems and the crossing number
- Richter, Thomassen
- 1995
(Show Context)
Citation Context ...s. The bound on cr(K n ) is known to be exact for ns12 (see, e.g., [106]). The bound on cr(K n;m ) is exact for ns6 [132] and for n = 7, ms10 [260]. The bound on cr(C n \Theta Cm ) is exact for n = 3 =-=[109, 175, 178]-=-, for n = 4 [82, 35], and for n = 5 [175, 224]. In each case the first reference (or two) gives the proof for m = n, while the last reference uses an induction argument to extend this for general m. N... |

24 |
Five-coloring maps on surfaces
- Thomassen
- 1993
(Show Context)
Citation Context ... graphs can be 4-colored. Can locally planar graphs be 4-colored? No, Fisk [85] constructed graphs of arbitrarily large face-width with chromatic number five. But four is close, as shown by Thomassen =-=[235]-=-. 36 Theorem 6.2 A graph embedded on the orientable surface of genus g with edge-width at least 2 14g+6 is 5-colorable. The proof involves cutting along a set of cycles to reduce to a planar graph, th... |

23 |
Generating the triangulations of the projective plane
- Barnette
- 1982
(Show Context)
Citation Context ...se was also shown directly by Malnic and Nedela [155] and by Gao, Richter, and Seymour [90]. Barnette [24] and independently Vitray [247] found the k-minimal graphs for the projective plane. Barnette =-=[25, 26, 27]-=- has also discussed various ways to generate triangulations, polyhedral, and closed 2-cell maps in simple surfaces. Randby (cf [200]) has shown that any (fw = k)-minimal embeddings in the projective p... |

23 |
On the crossing numbers of products of cycles and graphs of order four
- Beineke, Ringeisen
- 1980
(Show Context)
Citation Context ... known to be exact for ns12 (see, e.g., [106]). The bound on cr(K n;m ) is exact for ns6 [132] and for n = 7, ms10 [260]. The bound on cr(C n \Theta Cm ) is exact for n = 3 [109, 175, 178], for n = 4 =-=[82, 35]-=-, and for n = 5 [175, 224]. In each case the first reference (or two) gives the proof for m = n, while the last reference uses an induction argument to extend this for general m. Note that K 10 is the... |

23 |
Über das Problem der Nachbargebiete
- Heffter
(Show Context)
Citation Context ...torial Descriptions of Embeddings We need a convenient combinatorial way to describe an embedding. It is easiest to begin with an orientable surface. The following was implicit in the work of Heffter =-=[112]-=- with Edmonds [80] and Youngs [264] usually credited with being the first to (respectively) dualize and formalize the process. Fix a consistent orientation at each point on the surface, say anticlockw... |