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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