# Graph factorization

In graph theory, a **factor** of a graph *G* is a spanning subgraph, i.e., a subgraph that has the same vertex set as *G*. A ** k-factor** of a graph is a spanning

*k*-regular subgraph, and a

**partitions the edges of the graph into disjoint**

*k*-factorization*k*-factors. A graph

*G*is said to be

**if it admits a**

*k*-factorable*k*-factorization. In particular, a

**1-factor**is a perfect matching, and a 1-factorization of a

*k*-regular graph is an edge coloring with

*k*colors. A

**2-factor**is a collection of cycles that spans all vertices of the graph.

## Contents

## 1-factorization[edit]

Not every 1-factorable graph (ie, a graph which has a 1-factorization), has to be a regular graph. Additionally, not all regular graphs are 1-factorable. A *k*-regular graph is 1-factorable if it has chromatic index *k*; examples of such graphs include:

- Any regular bipartite graph.
^{[1]}Hall's marriage theorem can be used to show that a*k*-regular bipartite graph contains a perfect matching. One can then remove the perfect matching to obtain a (*k*− 1)-regular bipartite graph, and apply the same reasoning repeatedly. - Any complete graph with an even number of nodes (see below).
^{[2]}

However, there are also *k*-regular graphs that have chromatic index *k* + 1, and these graphs are not 1-factorable; examples of such graphs include:

- Any regular graph with an odd number of nodes.
- The Petersen graph.

### Complete graphs[edit]

A 1-factorization of a complete graph corresponds to pairings in a round-robin tournament. The 1-factorization of complete graphs is a special case of Baranyai's theorem concerning the 1-factorization of complete hypergraphs.

One method for constructing a 1-factorization of a complete graph on an even number of vertices involves placing all but one of the vertices on a circle, forming a regular polygon, with the remaining vertex at the center of the circle. With this arrangement of vertices, one way of constructing a 1-factor of the graph is to choose an edge *e* from the center to a single polygon vertex together with all possible edges that lie on lines perpendicular to *e*. The 1-factors that can be constructed in this way form a 1-factorization of the graph.

The number of distinct 1-factorizations of *K*_{2}, *K*_{4}, *K*_{6}, *K*_{8}, ... is 1, 1, 6, 6240, 1225566720, 252282619805368320, 98758655816833727741338583040, ... OEIS: A000438.

### 1-factorization conjecture[edit]

Let *G* be a *k*-regular graph with 2*n* nodes. If *k* is sufficiently large, it is known that *G* has to be 1-factorable:

- If
*k*= 2*n*− 1, then*G*is the complete graph*K*_{2n}, and hence 1-factorable (see above). - If
*k*= 2*n*− 2, then*G*can be constructed by removing a perfect matching from*K*_{2n}. Again,*G*is 1-factorable. - Chetwynd & Hilton (1985) show that if
*k*≥ 12n/7, then*G*is 1-factorable.

The **1-factorization conjecture**^{[3]} is a long-standing conjecture that states that *k* ≈ *n* is sufficient. In precise terms, the conjecture is:

- If
*n*is odd and*k*≥*n*, then*G*is 1-factorable. If*n*is even and*k*≥*n*− 1 then*G*is 1-factorable.

The overfull conjecture implies the 1-factorization conjecture.

### Perfect 1-factorization[edit]

A **perfect pair** from a 1-factorization is a pair of 1-factors whose union induces a Hamiltonian cycle.

A **perfect 1-factorization** (P1F) of a graph is a 1-factorization having the property that every pair of 1-factors is a perfect pair. A perfect 1-factorization should not be confused with a perfect matching (also called a 1-factor).

In 1964, Anton Kotzig conjectured that every complete graph *K*_{2n} where *n* ≥ 2 has a perfect 1-factorization. So far, it is known that the following graphs have a perfect 1-factorization:^{[4]}

- the infinite family of complete graphs
*K*_{2p}where*p*is an odd prime (by Anderson and also Nakamura, independently), - the infinite family of complete graphs
*K*_{p + 1}where*p*is an odd prime, - and sporadic additional results, including
*K*_{2n}where 2*n*∈ {16, 28, 36, 40, 50, 126, 170, 244, 344, 730, 1332, 1370, 1850, 2198, 3126, 6860, 12168, 16808, 29792}. Some newer results are collected here.

If the complete graph *K*_{n + 1} has a perfect 1-factorization, then the complete bipartite graph *K*_{n,n} also has a perfect 1-factorization.^{[5]}

## 2-factorization[edit]

If a graph is 2-factorable, then it has to be 2*k*-regular for some integer *k*. Julius Petersen showed in 1891 that this necessary condition is also sufficient: any 2*k*-regular graph is 2-factorable.^{[6]}

If a connected graph is 2*k*-regular and has an even number of edges it may also be *k*-factored, by choosing each of the two factors to be an alternating subset of the edges of an Euler tour.^{[7]} This applies only to connected graphs; disconnected counterexamples include disjoint unions of odd cycles, or of copies of *K*_{2k+1}.

The Oberwolfach problem concerns the existence of 2-factorizations of complete graphs into isomorphic subgraphs. It asks for which subgraphs this is possible. This is known when the subgraph is connected (in which case it is a Hamiltonian cycle and this special case is the problem of Hamiltonian decomposition) but the general case remains unsolved.

## Notes[edit]

**^**Harary (1969), Theorem 9.2, p. 85. Diestel (2005), Corollary 2.1.3, p. 37.**^**Harary (1969), Theorem 9.1, p. 85.**^**Chetwynd & Hilton (1985). Niessen (1994). Perkovic & Reed (1997). West.**^**Wallis, W. D. (1997), "16. Perfect Factorizations",*One-factorizations*, Mathematics and Its Applications,**390**(1 ed.), Springer US, p. 125, doi:10.1007/978-1-4757-2564-3_16, ISBN 978-0-7923-4323-3**^**Bryant, Darryn; Maenhaut, Barbara M.; Wanless, Ian M. (May 2002), "A Family of Perfect Factorisations of Complete Bipartite Graphs",*Journal of Combinatorial Theory*, A,**98**(2): 328–342, doi:10.1006/jcta.2001.3240, ISSN 0097-3165**^**Petersen (1891), §9, p. 200. Harary (1969), Theorem 9.9, p. 90. See Diestel (2005), Corollary 2.1.5, p. 39 for a proof.**^**Petersen (1891), §6, p. 198.

## References[edit]

- Bondy, John Adrian; Murty, U. S. R. (1976),
*Graph Theory with Applications*, North-Holland, ISBN 0-444-19451-7, archived from the original on 2012-06-16, Section 5.1: "Matchings". - Chetwynd, A. G.; Hilton, A. J. W. (1985), "Regular graphs of high degree are 1-factorizable",
*Proceedings of the London Mathematical Society*,**50**(2): 193–206, doi:10.1112/plms/s3-50.2.193. - Diestel, Reinhard (2005),
*Graph Theory*(3rd ed.), Springer, ISBN 3-540-26182-6, Chapter 2: "Matching, covering and packing". Electronic edition. - Harary, Frank (1969),
*Graph Theory*, Addison-Wesley, ISBN 0-201-02787-9, Chapter 9: "Factorization". - Hazewinkel, Michiel, ed. (2001) [1994], "One-factorization",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Niessen, Thomas (1994), "How to find overfull subgraphs in graphs with large maximum degree",
*Discrete Applied Mathematics*,**51**(1–2): 117–125, doi:10.1016/0166-218X(94)90101-5. - Perkovic, L.; Reed, B. (1997), "Edge coloring regular graphs of high degree",
*Discrete Mathematics*, 165–166: 567–578, doi:10.1016/S0012-365X(96)00202-6. - Petersen, Julius (1891), "Die Theorie der regulären graphs",
*Acta Mathematica*,**15**: 193–220, doi:10.1007/BF02392606. - West, Douglas B. "1-Factorization Conjecture (1985?)".
*Open Problems – Graph Theory and Combinatorics*. Retrieved 2010-01-09. - Weisstein, Eric W. "Graph Factor".
*MathWorld*. - Weisstein, Eric W. "k-Factor".
*MathWorld*. - Weisstein, Eric W. "k-Factorable Graph".
*MathWorld*.

## Further reading[edit]

- Plummer, Michael D. (2007), "Graph factors and factorization: 1985–2003: A survey",
*Discrete Mathematics*,**307**(7–8): 791–821, doi:10.1016/j.disc.2005.11.059.