#7365: Petersen's 2-factor theorem
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Reporter: ncohen | Owner: rlm
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.3
Component: graph theory | Keywords:
Work_issues: | Author:
Reviewer: | Merged:
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Changes (by ncohen):
* status: new => needs_review
* milestone: => sage-4.3
Old description:
> Implement a function corresponding to Petersen's theorem on 2-factors.
>
> This theorem says that any 2r-regular graphs can be decomposed in
> 2-factors. If the graph is not regular and is of maximum degree Delta,
> then it can be decomposed as an union of Delta/2 disjoints (<=2)-factors.
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 ).
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
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7365#comment:1>
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