#7365: Petersen's 2-factor theorem
----------------------------+-----------------------------------------------
Reporter: ncohen | Owner: rlm
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.3.1
Component: graph theory | Keywords:
Work_issues: | Author:
Upstream: N/A | Reviewer:
Merged: |
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Changes (by ncohen):
* status: needs_work => needs_review
Old description:
> As the docstring says :
>
> Petersen's 2-factor decomposition theorem asserts that any
> `2r`-regular graph `G` can be decomposed into 2-factors.
> Equivalently, it means that the edges of any `2r`-regular
> graphs can be partitionned in `r` sets `C_1,\dots,C_r` such
> that for all `i`, the set `C_i` is a disjoint union of cycles
> ( a 2-regular graph ).
>
> As any graph of maximal degree `\Delta` can be completed into
> a regular graph of degree `2\lceil\frac\Delta 2\rceil`, this
> result also means that the edges of any graph of degree `\Delta`
> can be partitionned in `r=2\lceil\frac\Delta 2\rceil` sets
> `C_1,\dots,C_r` such that for all `i`, the set `C_i` is a
> graph of maximal degree 2 ( a disjoint union of paths
> and cycles ).
>
> This patch both creates a new file in the graph directory, named
> graph_decomposition ( which will very soon contain many functions, do
> not worry about it !! ) into which is defined the function
> two_factor_petersen.
>
> As the moment, this patch requires many others which have not been merged
> :
> * #6679 Edge coloring function
> * #7270 Linear Programming class
> * #7268 or #7333 as a LP solver
> * #7364 eulerian orientation of a graph
>
> Perhaps the best thing to do is to review these patches before this very
> one.
>
> Nathann
New description:
As the docstring says :
Petersen's 2-factor decomposition theorem asserts that any
`2r`-regular graph `G` can be decomposed into 2-factors.
Equivalently, it means that the edges of any `2r`-regular
graphs can be partitionned in `r` sets `C_1,\dots,C_r` such
that for all `i`, the set `C_i` is a disjoint union of cycles
( a 2-regular graph ).
As any graph of maximal degree `\Delta` can be completed into
a regular graph of degree `2\lceil\frac\Delta 2\rceil`, this
result also means that the edges of any graph of degree `\Delta`
can be partitionned in `r=2\lceil\frac\Delta 2\rceil` sets
`C_1,\dots,C_r` such that for all `i`, the set `C_i` is a
graph of maximal degree 2 ( a disjoint union of paths
and cycles ).
Nathann
--
Comment:
Hello !!!
As mentionned, I moved this function toward graph.py, and will patiently
wait for the splitting of graph.py to begin creating new files.. :-)
(please do not forget to install GLPK or CBC before testing it because of
#7734)
Nathann
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7365#comment:3>
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