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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]
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 ...
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 ...
The Meredith graph, a quartic graph with 70 vertices that is 4-connected but has no Hamiltonian cycle, disproving a conjecture of Crispin Nash-Williams. [ 4 ] Every medial graph is a quartic plane graph , and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs. [ 5 ]
Since there is no way of partitioning the faces into two subsets that produce a sum obeying Grinberg's theorem, there can be no Hamiltonian cycle. [1] For instance, for the graph in the figure, all the bounded faces have 5 or 8 sides, but the unbounded face has 9 sides, so it satisfies this condition on numbers of sides and is not Hamiltonian.
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 Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by Kempe (1886). The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no 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.