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This identity's restriction to the case when is unitary, + + = + and =, where is the identity m×m-matrix, makes the (2m+n)×(2m+n)-matrix in the equality's right side unitary and its Hamiltonian cycle polynomial computable, hence, in polynomial time what therefore generalizes the above-given formula for the Hamiltonian cycle polynomial of a ...
A graph that can be proven non-Hamiltonian using Grinberg's theorem. In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian.
At about the same time as Hamilton, Thomas Kirkman in England was also studying Hamiltonian cycles on polyhedra. [17] Hamilton visited Kirkman in 1861, and presented him with a copy of the icosian game. [10] Despite this related work, some of which was much earlier, Hamiltonian cycles came to be named for Hamilton and for his work on the ...
A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time.
The dodecahedral graph is Hamiltonian, meaning a path visits all of its vertices exactly once. The name of this property is named after William Rowan Hamilton, who invented a mathematical game known as the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron. [38]
Dirac's theorem on Hamiltonian cycles, the statement that an n-vertex graph in which each vertex has degree at least n/2 must have a Hamiltonian cycle Dirac's theorem on chordal graphs , the characterization of chordal graphs as graphs in which all minimal separators are cliques
Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small ...
A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...