Hamiltonian Cycle

A Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once (Skiena 1990, p. 196). A graph possessing a Hamiltonian cycle is said to be a Hamiltonian graph. By convention, the singleton graph  is considered to be Hamiltonian even though it does not posses a Hamiltonian cycle, while the connected graph on two nodes  is not.

The Hamiltonian cycle is named after Sir William Rowan Hamilton, who devised a puzzle in which such a path along the polyhedron edges of an dodecahedron was sought (the Icosian game).

A Hamiltonian cycle of a graph can be computed efficiently in the Wolfram Language using FindHamiltonianCycle[g][[All, All, 1]][[1]] (where the cycle returned is not necessarily the lexicographically first one). All simple (undirected) cycles of a graph can be computed time-efficiently (but with a memory overhead of more than 10 times that needed to represent the actual cycles) using Sort[FindHamiltonianCycle[g, All][[All, All, 1]]]. (Note the cycles returned are not necessarily returned in sorted order by default.) Possible Method options to FindHamiltonianCycle include "Backtrack", "Heuristic", "AngluinValiant", "Martello", and "MultiPath". In addition, the Wolfram Language command FindShortestTour[g] attempts to find a shortest tour, which is a Hamiltonian cycle (with initial vertex repeated at the end) for a Hamiltonian graph  if it returns a list with first element equal to the vertex count of .

Precomputed lists of Hamiltonian cycles for many named graphs can be obtained using GraphData[graph, "HamiltonianCycles"]. Precomputed counts of the corresponding number of Hamiltonian cycles may similarly be obtained using GraphData[graph, "HamiltonianCycleCount"]..

The total numbers of directed Hamiltonian cycles for all simple graphs of orders , 2, ... are 0, 0, 2, 10, 58, 616, 9932, 333386, 25153932, 4548577688, ... (OEIS A124964).

In general, the problem of finding a Hamiltonian cycle is NP-complete (Karp 1972; Garey and Johnson 1983, p. 199), so the only known way to determine whether a given general graph has a Hamiltonian cycle is to undertake an exhaustive search. Rubin (1974) describes an efficient search procedure that can find some or all Hamilton paths and circuits in a graph using deductions that greatly reduce backtracking and guesswork. A probabilistic algorithm due to Angluin and Valiant (1979), described by Wilf (1994), can also be useful to find Hamiltonian cycles and paths.

All Platonic solids are Hamiltonian (Gardner 1957), as illustrated above.

Khomenko and Golovko (1972) gave a formula giving the number of graph cycles of any length, but its computation requires computing and performing matrix operations involving all subsets up to size , making it computationally expensive. A greatly simplified and improved version of the Khomenko and Golovko formula for the special case of -cycles (i.e., Hamiltonian cycles) gives

where  is the th matrix power of the submatrix of the adjacency matrix  with the subset  of rows and columns deleted (Perepechko and Voropaev).

The following table summarizes the numbers of (undirected) Hamiltonian cycles on various classes of graphs. The -hypercube is considered by Gardner (1986, pp. 23-24), who however gives the counts for an -hypercube for , 2, ... as 2, 8, 96, 43008, ... (OEIS A006069) which must be divided by  to get the number of distinct (directed) cycles counting shifts of points as equivalent regardless of starting vertex.

graph	OEIS	sequence
Andrásfai graph	A307902	0, 1, 5, 145, 8697, 1109389, 236702901, ...
antiprism graph	A306447	X, X, 16, 29, 56, 110, 225, 469, 991, 2110, 4511, ...
-black bishop graph	A307920	X, X, 0, 4, 704, 553008, , 13802629632, 1782158930138112, ...
cocktail party graph	A307923	0, 1, 16, 744, 56256, ...
complete graph	A001710	0, 0, 1, 3, 12, 60, 360, 2520, 20160, 181440, ...
complete bipartite graph	A010796	0, 1, 6, 72, 1440, 43200, 1814400, ...
complete tripartite graph	A307924	1, 16, 1584, 463104, 29928960, ...
-crossed prism graph	A007283	X, X, X, 6, 12, 24, 48, 96, 192, 384, 768, 1536, ...
crown graph	A306496	1, 6, 156, 4800, 208440, 11939760, 874681920, ...
cube-connected cycle	A000000	X, X, 6, 28628, ...
cycle graph	A000012	X, X, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
folded cube graph	A307925	X, 0, 3, 72, 23760, 332012113920, ...
grid graph	A003763	0, 1, 0, 6, 0, 1072, 0, 4638576, 0, ...
grid graph	A000000	0, 6, 0, ?, 0, ...
halved cube graph	A307926	0, 0, 3, 744, 986959440, 312829871511322359060480, ...
hypercube graph	A066037	0, 1, 6, 1344, 906545760, ...
-king graph	A140519	X, 3, 16, 2830, 2462064, 22853860116, ...
-knight graph	A001230	X, 0, 0, 0, 0, 9862, 0, 13267364410532, ...
-ladder graph	A057427	0, 1, 1, 1, 1, 1, 1, ...
Möbius ladder	A103889	X, X, X, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, ...
Mycielski graph	A307927	0, 1, 10, 102310, ...
odd graph	A301557	X, 1, 0, 1419264, ...
permutation star graph	A000000	0, 0, 1, 18, ...
prism graph	A103889	X, X, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, ...
-queen graph	A307928	0, 3, 1960, 402364270, 39741746126749664, ...
rook graph	A269561	X, 1, 48, 284112, 167875338240, ...
sun graph	A000012	X, X, 1, 1, 1, 1, 1, 1, ...
torus grid graph	A222199	X, X, 48, 1344, 23580, 3273360, ...
transposition graph	A307896	0, 0, 6, 569868288, ...
triangular graph	A307930	X, 0, 1, 16, 3216, 9748992, ...
triangular grid graph	A112676	1, 1, 3, 26, 474, 17214, 685727, ...
wheel graph	A000027	X, X, X, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...
-white bishop graph	A307929	X, X, 1, 4, 396, 553008, 4701600128, 1782158930138112, ...

Closed forms for some of these classes of graphs are summarized in the following table, where , , and  are the roots of  and  is a modified Bessel function of the second kind.

graph	formula
antiprism graph
cocktail party graph
complete graph
complete bipartite graph
complete tripartite graph
-crossed prism graph
crown graph
cycle graph	1
Hanoi graph	1
ladder graph	1
Möbius ladder
prism graph
sun graph	1
wheel graph

REFERENCES

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