# hamiltonian: Finds a Hamiltonian path or cycle

## Description

A Hamiltionian path or cycle (a.k.a. Hamiltonian circuit) is a path through a graph that visits each vertex exactly once, resp. a closed path through the graph.

## Usage

`hamiltonian(edges, start = 1, cycle = TRUE)`

## Arguments

edges

an edge list describing an undirected graph.

start

vertex number to start the path or cycle.

cycle

Boolean, should a path or a full cycle be found.

## Value

Returns a vector containing vertex number of a valid path or cycle, or `NULL` if no path or cycle has been found (i.e., does not exist); If a cycle was requested, there exists an edge from the last to the first vertex in this list of edges.

## Details

`hamiltonian()` applies a backtracking algorithm that is relatively efficient for graphs of up to 30--40 vertices. The edge list is first transformed to a list where the `i`-th component contains the list of all vertices connected to vertex `i`.

The edge list must be of the form `c(v1, v2, v3, v2, ...)` meaning that there are edges `v1 --> v2, v3 --> v4`, etc., connecting these vertices. Therefore, an edge list has an even number of entries.

If the function returns `NULL`, there is no Hamiltonian path or cycle. The function does not check if the graph is connected or not. And if `cycle = TRUE` is used, then there also exists an edge from the last to the first entry in the resulting path.

Ifa Hamiltonian cycle exists in the graph it will be found whatever the starting vertex was. For a Hamiltonian path this is different and a successful search may very well depend on the start.

## References

Papadimitriou, Ch. H., and K. Steiglitz (1998). Optimization Problems: Algorithms and Complexity. Prentice-Hall/Dover Publications.

Package `igraph`

## Examples

Run this code
``````# NOT RUN {
## Dodekaeder graph
D20_edges <- c(
1,  2,  1,  5,  1,  6,  2,  3,  2,  8,  3,  4,  3, 10,  4,  5,  4, 12,
5, 14,  6,  7,  6, 15,  7,  8,  7, 16,  8,  9,  9, 10,  9, 17, 10, 11,
11, 12, 11, 18, 12, 13, 13, 14, 13, 19, 14, 15, 15, 20, 16, 17, 16, 20,
17, 18, 18, 19, 19, 20)
hamiltonian(D20_edges, cycle = TRUE)
#   1  2  3  4  5 14 13 12 11 10  9  8  7 16 17 18 19 20 15  6
hamiltonian(D20_edges, cycle = FALSE)
#   1  2  3  4  5 14 13 12 11 10  9  8  7  6 15 20 16 17 18 19

## Herschel graph
# The Herschel graph the smallest non-Hamiltonian polyhedral graph.
H11_edges <- c(
1,  2,  1,  8,  1,  9,  1, 10,  2,  3,  2, 11,  3,  4,  3,  9,  4,  5,
4, 11,  5,  6,  5,  9,  5, 10,  6,  7,  6, 11,  7,  8,  7, 10,  8, 11)
hamiltonian(H11_edges, cycle = FALSE)
# NULL

# }
# NOT RUN {
## Example: Graph constructed from squares
N <- 45  # 23, 32, 45
Q <- (2:trunc(sqrt(2*N-1)))^2
sq_edges <- c()
for (i in 1:(N-1)) {
for (j in (i+1):N) {
if ((i+j) <!-- %in% Q) { -->
sq_edges <- c(sq_edges, i, j)
}
}

require(igraph)
sq_graph <- make_graph(sq_edges, directed=FALSE)
plot(sq_graph)

if (N == 23) {
# does not find a path with start=1 ...
hamiltonian(sq_edges, start=18, cycle=FALSE)
# hamiltonian(sq_edges)                     # NULL
} else if (N == 32) {
# the first of these graphs that is Hamiltonian ...
# hamiltonian(sq_edges, cycle=FALSE)
hamiltonian(sq_edges)
} else if (N == 45) {
# takes much too long ...
# hamiltonian(sq_edges, cycle=FALSE)
hamiltonian(sq_edges)
}
# }
``````

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