# Seeking an English-language proof of “Petersen’s classic decomposition theorem”: $2r$-regular graphs can be decomposed into $r$ disjoint $2$-factors.

From this paper we find the quote:

Lemma 1 (Petersen [2]) Every $2r$-regular graph can be decomposed into $r$ disjoint $2$-factors.

They described it as "Petersen’s classic decomposition theorem about regular graphs of even degree". The reference [2] is:

J. Petersen, Die Theorie der Regularen Graphen, Acta Math. 15 (1891), 193-220.

I don't have access to this paper, nor would I think it would do me much good, since the title suggests it's not in English.

Question: Where else can I find an English language proof of this result?

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Here is a link to the article: J. Petersen, Die Theorie der regulären Graphs which is indeed in German. – Martin Jan 22 '13 at 8:23

I am not sure if this suits you since the disjoint condition is omitted but in

• Laszlo Lovász; Combinatorial Problems and Exercises (2nd edition). North-Holland, Amsterdam (page 61. exercise 40.)

there is the following exercise (with hints and proofs included)

Every $2r$ regular graph is the union of $r$ 2-factors.

I am not sure if the disjoint part of your claim is just part of a different definition of factors than the one used in Lovász or if there is a way to modify the proof there to show this additional condition.

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Hmm... I think you get the "disjoint" property for free -- if they weren't disjoint, the union wouldn't be a $2r$-regular graph. – Douglas S. Stones Jan 22 '13 at 12:08

You can find it in Reinhard Diestel's "Graph Theory" textbook (Corollary 2.1.5 in the 4th edition).

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