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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 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 ...
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.
There is always a Hamiltonian cycle in the wheel graph and there are + cycles in W n (sequence A002061 in the OEIS). The 7 cycles of the wheel graph W 4 . For odd values of n , W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color.
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]
If negative weights and negatively weighted cycles are allowed, then finding a minimum cycle basis (without restriction) is also NP-hard, as it can be used to find a Hamiltonian cycle: if a graph is Hamiltonian, and all edges are given weight −1, then a minimum weight cycle basis necessarily includes at least one Hamiltonian cycle.
A Hamiltonian cycle on a tesseract with vertices labelled with a 4-bit cyclic Gray code Every hypercube Q n with n > 1 has a Hamiltonian cycle , a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path exists between two vertices u and v if and only if they have different colors in a 2 -coloring of the graph.
Illustration for the proof of Ore's theorem. In a graph with the Hamiltonian path v 1...v n but no Hamiltonian cycle, at most one of the two edges v 1 v i and v i − 1 v n (shown as blue dashed curves) can exist. For, if they both exist, then adding them to the path and removing the (red) edge v i − 1 v i would produce a Hamiltonian cycle.