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  |
 |
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