line_graph for multigraphs

[fabien-vignes]

We generalize the line_graph method to graphs possibly with multiple edges and/or loops.
It is useful in order to compute the line-graph of a multigraph but also to generate the edge-colorings of a multigraph from the all_graph_coloring method.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

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[fchapoton]

one failing doctest in

sage -t --long --warn-long 30.0 --random-seed=286735480429121101562228604801325644303 src/sage/graphs/generic_graph.py

to be investigated

[dcoudert]

I have other possible changes in mind, to get a faster code and to avoid calls to hash, but we can do that a future PR. Let us first finalize this one.

A more important question is if we should directly label the vertices of the line-graph with integers instead of tuples.

[fabien-vignes]

one failing doctest in

sage -t --long --warn-long 30.0 --random-seed=286735480429121101562228604801325644303 src/sage/graphs/generic_graph.py

to be investigated

In constast with the current version of line_graph, the new proposed method makes a copy of the graph g given as an input. The vertices and edges of the copy are sorted. This makes the order of the output of coarsest_equitable_refinement in the failing doctest to be slightly different (but fundamentally the same).
A way to solve this would be to modify the example of coarsest_equitable_refinement (in sage/src/sage/graphs/generic_graph.py) as follows:

sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False)
sage: L = ss.coarsest_equitable_refinement(prt)
sage: sorted(sorted(parts) for parts in L)
[[(0, 1)], [(0, 2), (0, 4)], [(0, 3)], [(1, 2), (1, 4)], [(2, 3), (3, 4)]]

[fabien-vignes]

one failing doctest in
sage -t --long --warn-long 30.0 --random-seed=286735480429121101562228604801325644303 src/sage/graphs/generic_graph.py
to be investigated

In constast with the current version of line_graph, the new proposed method makes a copy of the graph g given as an input. The vertices and edges of the copy are sorted. This makes the order of the output of coarsest_equitable_refinement in the failing doctest to be slightly different (but fundamentally the same). A way to solve this would be to modify the example of coarsest_equitable_refinement (in sage/src/sage/graphs/generic_graph.py) as follows:

sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False) sage: L = ss.coarsest_equitable_refinement(prt) sage: sorted(sorted(parts) for parts in L) [[(0, 1)], [(0, 2), (0, 4)], [(0, 3)], [(1, 2), (1, 4)], [(2, 3), (3, 4)]]

With the modifications proposed by @dcoudert, the PR should pass the doctests without modification.

[dcoudert]

Feel free to integrate all the changes I proposed.

Also, you may add the definition you use for the line graph of a multigraph. As we discussed, we have 2 cases: connect 2 vertices if the corresponding edges are incident in $g$, or connect 2 edges if the corresponding edges have exactly one extremity in common in $g$. You use the first definition. We may add the second later.

[dcoudert]

LGTM. The only remaining issue is that comments should be formatted in 80 columns mode (should node exceed the 80th columns of the text).
@fchapoton should we do the change now or let it for a future cleaning of the file (which will happen soon I think) ?

[fchapoton]

Given that this is the first Pull Request of Fabien, I suggest that this is approved right now.

[dcoudert]

LGTM.

Thank you for this improvement.