Hamiltonian Circuits. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected graph that visits each vertex. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle.Definitions · Properties · Bondy–Chvátal theorem · See also. Finding The Total Number Of Hamilton Circuits. The Traveling Salesman Problem is one of the. most intensely studied problems in. computational mathematics.

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Proof We only consider the case i.

Let C be an Eulerian circuit path hamiltonian circuits G and v an arbitrary vertex. Then each edge in C that hamiltonian circuits v must be followed by an edge in C that leaves v.

Thus the total number of edges incident at v must be even. Let Sn be the statement that connected graph of n vertices must be Eulerian if its every vertex has even degree.

### Hamiltonian path

Hence S1 is true because hamiltonian circuits Eulerian circuit can be obtained by traversing all loops if any one by one. Select a vertex v of G. The number of vertices must hamiltonian circuits doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.

Rahman-Kaykobad A simple graph with n vertices has a Hamiltonian path if, for every non-adjacent vertex pairs the sum of their degrees and their shortest path length is greater than n.

All Platonic solids are Hamiltonian circuits Gardneras illustrated above. Khomenko and Golovko gave a formula giving the number of graph cycles of hamiltonian circuits length, but its computation requires computing and performing matrix operations involving all subsets up to sizemaking it computationally expensive.

## Discrete Mathematics

A hamiltonian circuits simplified and improved version of the Khomenko and Hamiltonian circuits formula for the special case of -cycles i. The following table summarizes the numbers of undirected Hamiltonian cycles on various classes of graphs.

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Hamilton solved this problem using the Icosian Calculus, an algebraic structure based on roots of unity with many similarities to the quaternions also invented by Hamilton. This solution does not generalize hamiltonian circuits arbitrary graphs.

However, despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied hamiltonian circuits year earlier by Thomas Kirkman.