Nothing

```
zigzag <- function(n){
#Returns a matrix where each row is a hamiltonian path on complete graph on 1..n.
# Each pair (i,j) appears in at least one of the hamiltonians
m <- (n+1) %/% 2
a <- array(0,c(m,n))
a[1,1] <- 0
for (j in 2:n)
a[1,j] <- a[1,j-1] + (-1)^j*(j-1)
if (m >= 2) for (k in 2:m) a[k,] <- a[k-1,]+1
a <- a %% n +1
return(a)
}
close_path <- function(path){
n <- length(path)
if (path[1] != path[n])
path <- c(path,path[1])
return(path)
}
hpaths <- function(n, matrix=TRUE,cycle=NULL,...) {
#Returns a hamiltonian docomposition on the complete graph with n nodes
#If matrix is TRUE, returns a matrix where each row is a hamiltonian
#cotherwise concatenates the rows into a vector.
# If cycle is TRUE, the returned paths are cycles, the decomposition exists
# for odd n, but for even n the last row has some duplicate edges.
# If cycle is FALSE, the returned paths are open, the decomposition exists
# for even n, but for odd n the last row has some duplicate edges.
path1 <- NULL
if (length(n) !=1) {
path1 <- n
n <- length(path1)
}
if (is.null(cycle)) cycle <- n %% 2 != 0
if (cycle)
a <- cbind(1,zigzag(n-1) + 1,deparse.level=0)
else a <- zigzag(n)
if (!is.null(path1)) a <- permute_hpaths(path1,a)
if (!matrix) {
a <- as.vector(t(a))
if (cycle) a <- close_path(a)
}
return(a)
}
permute_hpaths <- function(path1,
paths= hpaths(length(path1)),
matrix=TRUE,...) {
# Permute the elements of paths so so path1 is the first row
#If matrix is TRUE, returns a matrix where each row is a hamiltonian
# path, otherwise concatenates the rows into a vector.
n <- length(path1)
o <- 1:n
o[paths[1,]] <- path1
paths <- matrix(o[paths],nrow=nrow(paths))
if (!matrix) {
paths <- as.vector(t(paths))
if (n%%2 != 0) paths <- close_path(paths)
}
return(paths)
}
```

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