ANTECEDENTES
2.3 Normatividad
In the previous chapter, we were introduced to Eulerian graphs, which are those graphs G possessing a circuit containing every edge of G. In this chapter, we turn our attention to those graphs G possessing a cycle containing every vertex of G.
6.1 Hamilton’s Icosian Game
William Rowan Hamilton (1805–1865) was gifted even as a child and his numer-ous interests and talents ranged from languages (having mastered many by age 10) to mathematics and physics. In 1832 he predicted that a ray of light passing through a biaxial crystal would be refracted into the shape of a cone. When this was experimentally confirmed, it was considered a major discovery and led to his being knighted in 1835, thereby becoming Sir William Rowan Hamilton.
Even today, Hamilton is regarded as one of the leading mathematicians and physicists of the 19th century.
Although Hamilton’s accomplishments were many, one of his best known in mathematics was his creation of a new algebraic system called quaternions, an extension of the complex numbers. On 16 October 1843, while walking with his wife along the Royal Canal in Dublin, Hamilton suddenly discovered a collection of 4-dimensional numbers a + bi + cj + dk, where a, b, c and d are real numbers, that formed a structure known as a division algebra. Furthermore,
i2= j2= k2= ijk = −1.
Hamilton carved these equations into the stone of the Brougham Bridge. In the quaternions, ij = k and ji = −k; so the quaternions are not commutative.
In 1856 Hamilton developed another example of a non-commutative alge-braic system in a game he called the Icosian Game, initially exhibited by Hamil-ton at a meeting of the British Association in Dublin. The Icosian Game (the prefix icos is from the Greek for twenty) consisted of a board on which were
placed twenty holes and some lines between certain pairs of holes. The diagram for this game is shown in Figure 6.1, where the holes are designated by the twenty consonants of the English alphabet.
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F Z P
B C
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Figure 6.1: Hamilton’s Icosian Game
Hamilton later sold the rights of his game for 25 pounds to John Jaques
& Son, a game manufacturer especially well known as a dealer in chess sets.
The preface to the instruction pamphlet for the Icosian Game, prepared by Hamilton for marketing the game in 1859, read as follows:
In this new Game (invented by Sir WILLIAM ROWAN HAMIL-TON, LL.D., & c., of Dublin, and by him named Icosian from a Greek word signifying ‘twenty’) a player is to place the whole or part of a set of twenty numbered pieces or men upon the points or in the holes of a board, represented by the diagram above drawn, in such a manner as always to proceed along the lines of the figure, and also to fulfill certain other conditions, which may in various ways be assigned by another player. Ingenuity and skill may thus be exer-cised in proposing as well as in resolving problems of the game. For example, the first of the two players may place the first five pieces in any five consecutive holes, and then require the second player to place the remaining fifteen men consecutively in such a manner that the succession may be cyclical, that is, so that No. 20 may be adja-cent to No. 1; and it is always possible to answer any question of this kind. Thus, if B C D F G be the five given initial points, it is allowed to complete the succession by following the alphabetical order of the twenty consonants, as suggested by the diagram itself;
but after placing the piece No. 6 in hole H, as above, it is also al-lowed (by the supposed conditions) to put No. 7 in X instead of J, and then to conclude with the succession, W R S T V J K L
6.1. HAMILTON’S ICOSIAN GAME 127 M N P Q Z. Other Examples of Icosian Problems, with solutions of some of them, will be found in the following page.
Another (traveler) version of Hamilton’s Icosian Game was labeled as NEW PUZZLE
TRAVELLER’S DODECAHEDRON or
A VOYAGE ROUND THE WORLD
In this game, the twenty vertices of the dodecahedron, labeled with the twenty consonants, stood for twenty cities of the world:
B. Brussels H. Hanover N. Naples T. Toholsk C. Canton J. Jeddo P. Paris V. Vienna D. Delhi K. Kashmere Q. Quebec W. Washington F. Frankfort L. London R. Rome X. Xenia G. Geneva M. Moscow S. Stockholm Z. Zanzibar
The idea was thus to construct a round trip around the world where each of the 20 cities would be visited on the trip exactly once.
Of course, the diagram of Hamilton’s Icosian game shown in Figure 6.1 can be immediately interpreted as a graph (see Figure 6.2), where the lines in the diagram are the edges of the graph and the holes are its vertices. Indeed, the graph of Figure 6.2 can be considered as the graph of the geometric solid called the dodecahedron (where the prefix dodec is from the Greek for twelve, pertaining to the twelve faces of the solid). This subject will be discussed in more detail in Chapter 10.
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Figure 6.2: The graph of the dodecahedron
6.2 Sufficient Conditions for Hamiltonicity
The problems proposed by Hamilton in his Icosian Game gave rise to concepts in graph theory, which eventually became a popular subject of study by math-ematicians. Let G be a graph. A path in G that contains every vertex of G is called a Hamiltonian path of G, while a cycle in G that contains every vertex of G is called a Hamiltonian cycle of G. A graph that contains a Hamiltonian cycle is itself called Hamiltonian. Certainly, the order of every Hamiltonian graph is at least 3 and every Hamiltonian graph contains a Hamiltonian path.
On the other hand, a graph with a Hamiltonian path need not be Hamiltonian.
The graph G1of Figure 6.3 is Hamiltonian and therefore contains both a Hamil-tonian cycle and a HamilHamil-tonian path. The graph G2 contains a Hamiltonian path but is not Hamiltonian; while G3contains neither a Hamiltonian cycle nor a Hamiltonian path.
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Figure 6.3: Hamiltonian paths and cycles in graphs
In 1855 (the year before Hamilton developed his Icosian Game) the Reverend Thomas Penyngton Kirkman (1806–1895) studied such questions as whether it is possible to visit all corners (vertices) of a polyhedron exactly once by moving along edges of the polyhedron and returning to the starting vertex. He observed that this could be done for some polyhedra but not all. While Kirkman had studied Hamiltonian cycles on general polyhedra and had preceded Hamilton’s work on the dodecahedron by several months, it is Hamilton’s name that became associated with spanning cycles in graphs, not Kirkman’s.