Topological map

Topological map is a term from the mathematical branch of topological graph theory . The term is particularly important in connection with studies of coloring problems and here not least in connection with the four-color theorem and related mathematical theorems .

Definitions and terms

• A topological map on a surface is a triple , where and two are finite set systems of subsets of and also represent a finite set. ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle F\subset \mathbb {R} ^{d}\;(d{\text{ ganzzahlig}}\;,\;d\geq 2)}$ ${\displaystyle {\mathfrak {K}}=({\mathfrak {L}},{\mathfrak {G}},E)}$ ${\displaystyle {\mathfrak {L}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle F}$ ${\displaystyle E\subset F}$
  • It is the edge set of a topological graph in and the associated set of nodes . ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle F}$ ${\displaystyle E}$
  • ${\displaystyle E}$consists of exactly those points of which occur as the start or end point for one of the Jordan curves . ${\displaystyle F}$ ${\displaystyle g\in {\mathfrak {G}}}$
  • The set system consists precisely of the path-related components of the complement set . ${\displaystyle {\mathfrak {L}}}$ ${\displaystyle F\setminus \bigcup {\mathfrak {G}}}$
• Here, each element of is referred to as land , each element of as boundary line and each element of as corner of the topological map . ${\displaystyle {\mathfrak {L}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle E}$ ${\displaystyle {\mathfrak {K}}}$
• A point is the edge point of a country belonging to the map if it belongs to the relative topological closure of in . ${\displaystyle x\in \mathbb {R} ^{d}}$ ${\displaystyle L}$ ${\displaystyle {\overline {L}}\cap F}$ ${\displaystyle L}$ ${\displaystyle F}$
• Two countries and of are called neighboring or neighboring countries if one occurs under the border lines of which consists entirely of boundary points both from and from . ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$ ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$
• A mapping given to an integer is called a coloration . ${\displaystyle n>0}$ ${\displaystyle f\colon \,{\mathfrak {L}}\to \{1,\ldots ,n\}}$ ${\displaystyle n}$
  • The elements of are called colors (in accordance with the conventions of graph theory ) . ${\displaystyle \{1,\ldots ,n\}}$
  • A staining is permitted if each two neighboring countries virtue always two different colors are assigned. ${\displaystyle n}$ ${\displaystyle f\colon \,{\mathfrak {L}}\to \{1,\ldots ,n\}}$ ${\displaystyle f}$
  • Allows a topological map on an integer a permissible staining , but not permitted coloring with less than color, we call this integer the chromatic number of and is denoted by . ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle F}$ ${\displaystyle n>0}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle \chi ({\mathfrak {K}})}$
  • If one forms over all topological maps on ${\displaystyle F}$ the supremum of   ${\displaystyle \sup\{\chi ({\mathfrak {K}}):{\mathfrak {K}}\;{\text{topologische Landkarte auf F}}\}}$   all associated chromatic numbers and this is an integer , then this is the chromatic number of . It is designated with . ${\displaystyle <\infty }$ ${\displaystyle F}$ ${\displaystyle \chi (F)}$

annotation

• The areas examined are usually areas . ${\displaystyle F\subset \mathbb {R} ^{d}\;(d=2,3,4)}$

Significant tenets

• Weiske's theorem : If the one or more spheres are unified , there is no map with five countries neighboring in pairs. ${\displaystyle F}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle S^{2}}$
• Two-color theorem : If a rectangle and the boundary lines of a given topological map are such that each of them only runs between the edge points of the rectangle or represents a closed Jordan curve running within the rectangle , then there is always a permissible coloration for such a topological map . ${\displaystyle F}$ ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle {\mathfrak {K}}}$ ${\displaystyle 2}$
• Four-color theorem : Ifthe oneor more unit spheres, then there isa permissiblecolorfor every topological map. ${\displaystyle F}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle F}$ ${\displaystyle 4}$
• Five-color theorem : Ifthe oneor more unit spheres, then there isa permissiblecolorationfor every topological map. ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle F}$ ${\displaystyle 5}$
• Six-color set for the Möbius strip : The Möbius strip has the chromatic number and every topological map has a permissible coloration on it, whereby there is also at least one of these that does not get by with five colors. ${\displaystyle M}$ ${\displaystyle {\chi }(M)=6}$ ${\displaystyle 6}$
• Heawood's inequality : Ifa closed, orientable surface of gender is for an integerand is included, then there isa permissiblecolorationfor every topological map. In other words: for every such areathe chromatic number satisfiesthe inequality . ${\displaystyle p>0}$   ${\displaystyle H_{p}}$ ${\displaystyle p}$ ${\displaystyle m_{p}=\lfloor {\frac {7+{\sqrt {1+48p}}}{2}}\rfloor }$ ${\displaystyle H_{p}}$ ${\displaystyle m_{p}}$ ${\displaystyle H_{p}}$ ${\displaystyle {\chi }(H_{p})}$ ${\displaystyle {\chi }(H_{p})\leq m_{p}}$
  • The identity equation always applies to such a thing . ${\displaystyle H_{p}}$ ${\displaystyle {\chi }(H_{p})=m_{p}}$
  • In particular, the full torus has the chromatic number and every topological map has a permissible coloration on it, including those that do not get by with six colors. ${\displaystyle H_{1}}$ ${\displaystyle {\chi }(H_{1})=7}$ ${\displaystyle 7}$

Notes on the theorems

• Weiske's theorem goes back to the philologist Benjamin Gotthold Weiske (1783–1836), a friend of the mathematician August Ferdinand Möbius . The authorship of this result was found out by the geometer Richard Baltzer while looking through the Möbius estate. When Baltzer reported on Weiske's theorem and its history in a lecture in 1885, however, he then brought into the world the mistake that with Weiske's theorem, the four-color theorem was an easy inference. On the other hand, it is correct that, in contrast to the representation of Baltzer Weiske's theorem, alone enables an easy derivation of the five-color theorem. Baltzer's mistake was finally eliminated in 1959 by the geometer Harold Scott MacDonald Coxeter .
• The four-color theorem is regarded today by many, but by no means by all mathematicians as proven.
• Percy John Heawood already suspected that in Heawood's inequality even the equal sign has to apply and was finally proven in 1968 by the two mathematicians Gerhard Ringel and John William Theodore Youngs .

The thread problem

The thread problem consists in the question of how to solve the following task:

If possible, the smallest natural number is to be determined for a given natural number for which any selected different points on a closed, orientable surface of the gender can always be connected in pairs by simple Jordan curves in such a way that all these Jordan curves never cross one another and meet at most in the selected points . ${\displaystyle m\geq 3}$ ${\displaystyle \gamma (m)}$ ${\displaystyle H_{\gamma (m)}}$ ${\displaystyle \gamma (m)}$ ${\displaystyle p_{1},p_{2},\ldots ,p_{m}\in H_{\gamma (m)}}$ ${\displaystyle p_{1},p_{2},\ldots ,p_{m}}$

As it turns out, the thread problem can be solved and this results in the formula

${\displaystyle \gamma (m)=\lceil {\frac {(m-3)(m-4)}{12}}\rceil \;(m\in \mathbb {N} \;,\;m\geq 3)}$.

It also shows that the validity of this formula also implies the validity of Heawood's identity equation . ${\displaystyle {\chi }(H_{p})=m_{p}\;(p\in \mathbb {N} \;,\;p>0)}$

literature

• Rudolf Fritsch : The four-color set. History, topological foundations and idea of ​​proof . With the collaboration of Gerda Fritsch . BI Wissenschaftsverlag , Mannheim, Leipzig, Vienna, Zurich 1994, ISBN 3-411-15141-2 ( MR1270673 ).
• Konrad Jacobs , Dieter Jungnickel : Introduction to combinatorics (=  de Gruyter textbook ). 2nd, completely revised and expanded edition. de Gruyter , Berlin (et al.) 2004, ISBN 3-11-016727-1 .
• KP Müller , H. Wölpert : Illustrative topology . An introduction to elementary topology and graph theory (=  mathematics for teacher training ). BG Teubner Verlag , Stuttgart 1976, ISBN 3-519-02709-7 .
• Gerhard Ringel : The card coloring problem . In: Selecta Mathematica III (Ed. Konrad Jacobs) (=  Heidelberger Taschenbücher . Volume 86 ). Springer Verlag , Berlin, Heidelberg, New York 1971, ISBN 3-540-05333-6 ( MR0543809 ).
• Lutz Volkmann : Foundations of the graph theory . Springer Verlag, Vienna, New York 1996, ISBN 3-211-82774-9 ( MR1392955 ).
• Klaus Wagner : Theory of graphs (=  BI university pocket books. 248 / 248a). Bibliographisches Institut , Mannheim (inter alia) 1970, ISBN 3-411-00248-4 .

Individual evidence

1. ↑ The chromatic number is to be distinguished from the Euler characteristic of , although the same Greek letter appears as an identifier for both numbers. ${\displaystyle \chi (F)}$ ${\displaystyle F}$
2. ↑ ^{a b} Rudolf Fritsch: The four-color set. 1994, pp. 25-26, 128
3. ^ KP Müller, H. Wölpert: Illustrative topology. 1976, p. 67
4. ↑ Müller, Wölpert, op. Cit., Pp. 148-149
5. ↑ is the Gaussian bracket function . ${\displaystyle \lfloor {\cdot }\rfloor }$
6. ↑ Gerhard Ringel: The card coloring problem. in: Selecta Mathematica III 1971, pp. 30 ff., 45-47
7. ↑ ^{a b} Ringel, op.cit., P. 31
8. ↑ See four-color theorem # Criticism !
9. ↑ Ringel, op.cit., P. 32
10. ↑ is the rounding function . ${\displaystyle \lceil \cdot \rceil }$
11. ↑ Ringel, op. Cit., Pp. 32-33