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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.
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 ...
In all other cases it has a Hamiltonian cycle. [6] When n is congruent to 3 modulo 6 G(n, 2) has exactly three Hamiltonian cycles. [7] For G(n, 2), the number of Hamiltonian cycles can be computed by a formula that depends on the congruence class of n modulo 6 and involves the Fibonacci numbers. [8]
It implies that computing, up to the -th power of , the Hamiltonian cycle polynomial of a unitary n×n-matrix over the infinite extension of any ring of characteristic q (not necessarily prime) by the formal variable is a # P-complete problem if isn't 2 and computing the Hamiltonian cycle polynomial of a -semi-unitary matrix (i.e. an n×n ...
Pages in category "Hamiltonian paths and cycles" The following 23 pages are in this category, out of 23 total. This list may not reflect recent changes. ...
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.
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.
The top two edges in the inner cycle must be in C, but this completes a non-spanning cycle, which cannot be part of a Hamiltonian cycle. Alternatively, we can also describe the ten-vertex 3-regular graphs that do have a Hamiltonian cycle and show that none of them is the Petersen graph, by finding a cycle in each of them that is shorter than ...