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Representation of Graphs in Diagrams of Graph Theory

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Visual Reasoning with Diagrams

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

The treatise of Dénes Kőnig Theorie der endlichen und unendlichen Graphen (1936) includes many diagrams. The style of the diagrams is typical of present-day texts of graph theory. In this treatise, many mathematical recreation problems are treated. Some of the problems were already treated by precedent mathematicians. In some of these early works, the diagrams were given, but the styles were not the same as that of Kőnig. Moreover, the way to use the diagrams in the early works is different from that of Kőnig.

Examining the diagrams, we will find that a certain type of diagrams became gradually influential. This historical aspect may be related to the formation of concepts of graph, but it will not be discussed here.

We will argue that Tarry’s talk “Géométrie de situation: nombre de manières distinctes de parcourir en une seule course toutes les allées d’un labyrinthe rentrant, en ne passant qu’une seule fois par chacune des allées” (1886) played an important role in the way to use the diagrams.

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Notes

  1. 1.

    I examined this relationship in my thesis [51], Chap. 6.

  2. 2.

    Kőnig made here a reference to the article by Sylvester entitled “On recent discoveries in mechanical conversion of motion” in 1873 [47]. This article treated a mode of producing motion in a straight line by a system of pure link-work without the aid of grooves or wheel-work, or any other means of constraint than that due to fixed centers, and joints for attaching or connecting rigid bars. Maybe here Kőnig had the following part of Sylvester’s article in mind: “The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation. [New paragraph] The sphere within which any theory of colligation works is not spatial but logical—such theory is concerned exclusively with the necessary laws of antecedence and consequence, or in one word of connection in the abstract, or in other words is a development of the doctrine of the compound parenthesis.” (The Collected Mathematical Papers of James Joseph Sylvester, vol. 3, pp. 23–24.)

  3. 3.

    Some of the collections in this genre were cited by Kőnig in his books on mathematical recreations in 1902 and 1905 [27, 28], and some of them were cited even in his treatise of 1936 [29]: Problèmes plaisans et délectables qui se font par les nombres by Claude-Gaspar Bachet de Méziriac [5–8] in the early 17th century; Récreations mathématiques et physiques by Jacques Ozanam [36, 37] in the 17th century; Récreations mathématiques (4 volumes) and L’Arithmétique amusante by Édouard Lucas [31–35] and Mathématiques et mathématiciens: pensées et curiosités by Alphonse Rebière [40, 41] in the 19th century; Mathematical Recreations and problems of past and present times (Mathematical Recreations and essays for the 4th edn. and later) by Walter William Rouse Ball [9–15] (and many later editions), Mathematische Mußestunden by Hermann Schubert [43–45] and Récréations arithmétiques and Curiosités géométriques by Émile Fourrey [24, 25] around 1900; Mathematische Unterhaltungen und Spiele and Mathematische Spiele by Wilhelm Ahrens [1–4] in the early 20th century. Also in the period close to the publication of Kőnig’s treatise of 1936, a book in this genre was published in Belgium: La mathématique des jeux ou récréations mathématiques by Maurice Kraitchik [30], which might suggest interest of mathematicians at that time in mathematical recreations.

    Anne-Marie Décaillot made historical researches on Lucas’ works [18–20].

    David Singmaster made a precise examination of all the editions of Ball’s book [46].

    Albrecht Heeffer [26] worked on a movement toward creation of this genre in the 17th century.

    In the UK and the USA in the 19th and the 20th century, amateur mathematicians or scientific writers—Samuel Loyd, Henry Ernst Dudeney, Martin Gardner and so on—also published in this domain. Some problems in their collections were taken from the collections listed above. This fact is interesting from the point of view of the popularization of mathematics, but, for the purpose of this article, we don’t need to examine precisely these works.

  4. 4.

    I translated these Hungarian books into English in my thesis [51].

  5. 5.

    As Décaillot says, “among the mathematical games and recreations of Euler, the traces of strings on the chessboard of Vandermonde in the 18th century, the ‘higher mathematics’ of Riemann and the Analysis situs of Poincaré, the geometry of situation has a fluctuating content which is structured only slowly during the 19th century” ([19], p. 129, my translation from French).

  6. 6.

    I examined the difference of the translations of Coupy and Lucas in Chap. 3 of my thesis [51].

  7. 7.

    A part of the line between A and B of Kőnig’s diagram is not connected, but it is only an error of printing. It should be continuous for consistency of the text.

  8. 8.

    Despite the fact that Poinsot speaks of a flexible string, he uses the length. This means that one looses the “geometry of situation”.

  9. 9.

    Poinsot mentioned Vandermonde [50] in the context of the problem of the knight’s move on a chessboard, as one of the problems concerning the “geometry of situation”. He meant that Vandermonde’s solution was simpler than Euler’s [23].

  10. 10.

    Charles-Ange Laisant (1841–1920) was a mathematician, and was a director of some reviews of mathematics.

  11. 11.

    It is unclear who the authors of the abstracts were, but someone in the bureau of the section which contained Tarry’s talk may have been the author: Président d’honneur: M. le Géneral FROLOW, major général du génie russe (Russian general major of engineering); Président: M. Ed. LUCAS, Prof. de math. spéciale au Lycée Saint-Louis (Professor of higher mathematics at the Saint-Louis High school); Vice-Président: M. Laisant, Député de la Seine, Anc. Él. de l’Éc. Polyt. (Deputy of the Seine, Alumnus of the Polytechnic School); Secrétaire: M. HEITZ, Él. de l’Éc. centr. des Arts et Manufact. (Student of the central school of Arts and Manufacture).

  12. 12.

    Tarry talked about figures of mazes according to his article in vol. 2 of the proceedings.

  13. 13.

    The problem treated by Tarry was therefore different from the problem treated by Poinsot, who described the possibility of tracing all the edges and diagonals of a polygon once and only once with one stroke.

  14. 14.

    Michel Reiss (1805–1869) was a mathematician from Frankfurt who worked mainly on the theory of determinants. He published an article about dominoes “Evaluation du nombre de combinaisons desquelles les 28 dés d’un jeu du domino sont susceptibles d’après la règle de ce jeu” (1871) of 58 pages [42].

  15. 15.

    That is, Édouard Lucas.

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  1. Equipe REHSEIS, Laboratoire SPHERE, Université Paris 7, 5 rue Thomas Mann, Case 7093, 75205, Paris Cedex 13, France

    Mitsuko Wate-Mizuno

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  1. IRIST, Université de Strasbourg, Rue de l'Université 7, Strasbourg, 67000, France

    Amirouche Moktefi

  2. Department of Philosophy, Yale University, Hillhouse Avenue 38, New Haven, 06511, Connecticut, USA

    Sun-Joo Shin

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Wate-Mizuno, M. (2013). Representation of Graphs in Diagrams of Graph Theory. In: Moktefi, A., Shin, SJ. (eds) Visual Reasoning with Diagrams. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0600-8_10

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