How to extract mutually disjoint perfect matchings?

[Phoenix]

Say I have a K_{n,n} and from it I want to extract d mutually disjoint perfect matchings.

How can this be done?

Hopefully the result can be obtained as a list of list of edges where each edge is given an integer 2-tuple corresponding to the vertices that this edge connects.

[Nathann Cohen]

Say I have a K_{n,n} and from it I want to extract d mutually disjoint perfect matchings.

How can this be done?

http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_coloring.html#sage.graphs.graph_coloring.edge_coloring 

[Phoenix]

Didn't get you. Can you explain a bit more?

[Vincent Delecroix]

If you follow the link then you can read

If vizing=True and value_only=False, return a partition of the edge set
into Δ+1 matchings.

Isn't that clear enough?

[Nathann Cohen]

> Didn't get you. Can you explain a bit more?

A partition of the edges of a graph into disjoint matchings is called
an edge-coloring. With the function I gave you, you can compute an
edge-coloring of your graph which, because that graph is a K_{n,n},
will be a collection of disjoint perfect matchings.

Note that it may be a bit slow. You have a faster way to obtain what
you desire with:

sage: n=5;designs.transversal_design(2,n,resolvable=True)._classes
[[[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]],
[[0, 6], [1, 7], [2, 8], [3, 9], [4, 5]],
[[0, 7], [1, 8], [2, 9], [3, 5], [4, 6]],
[[0, 8], [1, 9], [2, 5], [3, 6], [4, 7]],
[[0, 9], [1, 5], [2, 6], [3, 7], [4, 8]]]

What you see is a list of [list of pairs], and each [list of pairs] is
a perfect matching in graphs.CompleteBipartiteGraph(n,n). They are all
disjoint.

Nathann

[Phoenix]

Thanks.