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Is there a characterization of graphs whose edge set decomposes into a disjoint union of perfect matchings?

One trivial class of such graphs are $d$-regular $(n,n)$-bipartite graphs. Their edge set will decompose into $d$ disjoint perfect matchings.

David Richerby
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user6818
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1 Answers1

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Yes: we call such graphs 1-factorable (a 1-factor is also known as a perfect matching). All such graphs are regular, but the converse is not true. In fact, a $d$-regular graph $G$ is 1-factorable if and only if it is of class one, that is, $\chi'(G) = d$, where $\chi'(G)$ is the chromatic index of $G$.

Deciding if a $d$-regular graph is of class 1 is NP-complete (see e.g. [1]), so you likely cannot test this efficiently.

[1] Leven, Daniel, and Zvi Galil. "NP completeness of finding the chromatic index of regular graphs." Journal of Algorithms 4.1 (1983): 35-44.

Juho
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  • Thanks for the reply! (1) Do you have a reference for a proof of this NP-completeness? (2) Seems like there are other classes also? Any pedagogic reference for these? (3) Do you know if something special is known about the perfect matching polytope of such 1-factorable graphs? – user6818 Aug 14 '15 at 18:40
  • No, this is a characterization. That is to say, there are no other graph classes. The class of 1-factorable graphs is exactly the class of $d$-edge-colorable $d$-regular graphs. I don't think I know of anything better than what Wikipedia offers, see for instance here. – Juho Aug 14 '15 at 18:49