Construct eulerian paths on the complete graph where nodes are integers 1..n.

Constructs an eulerian on the complete graph where nodes are integers 1..n. The result in an euler tour for odd n. For even n the result is not exactly an euler tour or path because (n-2)/2 edges must be visited twice.

Usage

eseq(n)
eseqa(n)
kntour_drop(e)
kntour_add(e)

Arguments

n

a positive integer.

e

an euler tour on Kn where n is odd

Value

a numeric vector.

Details

The algorithm used for eseq builds up a path on 1..n by appending extra edges on to the path on nodes 1..(n-2).

The function eseqa constructs paths on 1..n using an alternative algorithm. For odd n, the tour starts at 1, then takes steps of size 1,2,..m repeatedly, where m is (n-1)/2, For even n, the path constructed is formed as eseqa(n+1), followed by dropping node n+1.

The function kntour_drop removes instances of n from the tour, creating an open approximately eulerian path on the complete graph with n-1 nodes.

The function kntour_add inserts an extra node n+1 into a tour on nodes 1, ..n. It adds a detour to the tour visiting all edges joining nodes 1..n to n+1. The result is an open approximately eulerian path on the complete graph with n+1 nodes.

References

see overview

Author

C.B. Hurley and R.W. Oldford

See also

hpaths, eulerian.

Examples

require(PairViz)
eseq(5)
#>  [1] 1 2 3 1 4 2 5 3 4 5 1
eseq(6)
#>  [1] 1 2 3 1 4 2 3 4 5 1 6 2 5 3 6 4 5 6