SIR WILLIAM ROWAN HAMILTON
It was not until Hamilton was 15 that his interests changed and he became excited about mathematics. The change was brought about when he met Zerah Colburn, the American lightning calculator, who, though only a youngster himself, gave a demonstration of his powers at an exhibition in Dublin. Shortly after, Hamilton read a copy of Newton's Universalls arithmetica through which Hamilton mastered analytic geometry and calculus. He then read the four volumes of the Principla and proceeded to the great works of the continent. Hamilton uncovered a mathematical error while reading LaPlace's Mechanique celeste and in 1823 wrote a paper on it that attracted considerable attention. In 1824 Hamilton entered Trinity College, Dublin. Hamilton's career at the university was unique, for in 1828, when he was only 21 years old and still an undergraduate, the electors unanimously appointed him Royal Astronomer of Ireland, Director of the Dunsink Observatory, and Professor of Astronomy at the University. He remained the Professor of Astronomy until his death in 1865. In 1833 he presented to the Irish Academy his significant paper in which the algebra of complex numbers appears as an algebra of ordered pairs of real numbers. In 1835 he was knighted.
Following his 1833 paper, Hamilton thought off and on for a long period of years on algebra of ordered triples and quadruples of real numbers. He always got stuck on how to define multiplications so as to preserve the familiar laws of that operation. Finally in 1843 while walking along the Royal Canal outside Dublin, it occurred to him that he had to sacrifice the commutative law, and the algebra of quaternions (a quaternion is an element of a system of four dimensional vectors obeying laws similar to complex numbers), the first noncumulative algebra, was suddenly born.
During the remaining twenty plus years of his life, Hamilton used most of his time and energy developing his quaternions, which he felt would be of revolutionary significance in mathematical physics. His great work, TreatL'se on Quaternions, appeared in 1853, after which he devoted himself to preparing an enlarged Elements of Quaternions, but he died in Dublin in 1865 before it was complete. His death was essentially from alcoholism and a generally rundown condition brought on by a very unhappy married life.
One of Hamilton's accomplishments was his work in graph theory. In 1859 Hamilton developed a game that he sold to a Dublin toy manufacturer. The game consisted of a wooden regular dodecahedron with the 20 corner points (vertices) labeled with the names of prominent cities. The objective of the game was to find a cycle along the edges of the solid so that each city was on the cycle exactly once. The figure below is the planar graph for this Platonic solid; such a cycle is designated by the darkened edges. This illustration leads us to the following definition
If G = (V,E) (where V = vertex set and E = edge set) is a graph or multigraph, we say that G has a Hamilton cycle if there is a cycle in G that contains every vertex in V. A Hamilton path is a path in G that contains each vertex.
Given a graph with a Hamilton cycle, we find that the deletion of any edge in the cycle results in a Hamilton path. It is possible, however, for a graph to have a Hamilton path without having a Hamilton cycle, as will be illustrated later.
It may seem that the existence of a Hamilton cycle (path) and the existence of an Euler circuit (trail) for a graph are similar problems. The Hamilton cycle (path) is designed to visit each vertex in a graph only once; the Euler circuit (trail) traverses the graph so that each edge is travelled exactly once. There is no helpful connection between the two ideas, and unlike the situation for Euler circuits (trails), there do not exist necessary and sufficient conditions on a graph G that guarantee the existence of a Hamilton cycle (path). If a graph has a Hamilton cycle, then it will at least be connected.
Example: If G is the graph in the figure below, the edges la,b), lb,c), ic,f), tfe), le,d), [d,g), (g,h) and th,i) yield a Hamilton path for G. But does G have a Hamilton cycle? a b c 9 h
Since G has nine vertices, if there is a Hamilton cycle in G it must contain nine edges. Let us start at vertex b and try to build a Hamilton cycle. With regard to the symmetry in the graph, it does not matter whether we go from b to c or to a. We will go to c. At c we can go to either f or to i. Using symmetry again, we go to f. Then we delete edge fc,i) from further consideration because we cannot return to vertex c. In order to include vertex i in our cycle, we must now go from f to i, and then to h to g. With edges fc,f) and (fi) in the cycle, we delete edge 16,f). But once we get to e we are stuck. Therefore, there is no Hamilton cycle for the graph, although there is a Hamilton path.
The degree of a vertex can be defined as follows: G is an undirected graph or multigraph. For any vertex v of G, the degree of v, written deg(v) is the number of edges in G that are incident with v.
For the graph shown below, deg(b) = deg(d) = deg(f) = deg(g) =2, deg(c) = 4, deg(e) = 0, and deg(h) = 1. For vertex a we have deg(a) = 3, because we count a loop twice. *e 9
We can now give a few helpful hints for trying to find a Hamilton cycle in graph G = (V,E):
[Ed. note] The work of both Euler and Hamilton did much to generate interest in the Theory of Graphs and enhance the use of graphs as mathematical models of all manner of networks. We n-dght regard this topic as one that had its birth in the realm of puzzles and games.
Robert H. Orr