Graph factorization

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Partition of a graph into spanning subgraphs

title: "Graph factorization" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["graph-theory-objects", "factorization"] description: "Partition of a graph into spanning subgraphs" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Graph_factorization" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Partition of a graph into spanning subgraphs ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/03/Desargues_graph_3color_edge.svg" caption="1-factor}}."] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a9/Petersen-graph-factors.svg" caption="1-factorable}}."] ::

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 k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2-factor is a collection of disjoint cycles that spans all vertices of the graph.

1-factorization

If a graph is 1-factorable then it has to be a regular graph. However, 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. 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). 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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bb/Complete-edge-coloring.svg" caption="1-factor}} consists of an edge from the center to a vertex of a [[heptagon]] together with all possible perpendicular edges"] ::

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 in a regular polygon, with the remaining vertex at the center. 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 K2, K4, K6, K8, ... is 1, 1, 6, 6240, 1225566720, 252282619805368320, 98758655816833727741338583040, ... ().

1-factorization conjecture

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

  • If k = 2n − 1, then G is the complete graph K2n, and hence 1-factorable (see above).
  • If k = 2n − 2, then G can be constructed by removing a perfect matching from K2n. Again, G is 1-factorable.
  • show that if k ≥ 12n/7, then G is 1-factorable. The 1-factorization conjecture 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. The conjecture was confirmed by Csaba, Kühn, Lo, Osthus and Treglown for sufficiently large n. | last1 = Csaba | first1 = Béla | last2 = Kühn | first2 = Daniela | last3 = Lo | first3 = Allan | last4 = Osthus | first4 = Deryk | last5 = Treglown | first5 = Andrew

| title = Proof of the 1-factorization and Hamilton decomposition conjectures | journal = Memoirs of the American Mathematical Society | date = June 2016 | doi = 10.1090/memo/1154

Perfect 1-factorization

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 K2n where n ≥ 2 has a perfect 1-factorization. So far, it is known that the following graphs have a perfect 1-factorization: | first = W. D. | last = Wallis | title = One-factorizations | publisher = Springer US | series = Mathematics and Its Applications | volume = 390 | edition = 1 | year = 1997 | chapter = 16. Perfect Factorizations | page = 125 | doi = 10.1007/978-1-4757-2564-3_16 | isbn = 978-0-7923-4323-3

  • the infinite family of complete graphs K2p 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 K2n where 2n ∈ {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. | last1 = Bryant | first1 = Darryn | last2 = Maenhaut | first2 = Barbara M. | last3 = Wanless | first3 = Ian M.

| title = A Family of Perfect Factorisations of Complete Bipartite Graphs | journal = Journal of Combinatorial Theory | series = A | volume = 98 | issue = 2 | pages = 328–342 | date = May 2002 | issn = 0097-3165 | doi = 10.1006/jcta.2001.3240 | doi-access= free

2-factorization

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

If a connected graph is 2k-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. This applies only to connected graphs; disconnected counterexamples include disjoint unions of odd cycles, or of copies of K2k+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 open.

References

Bibliography

  • {{citation |last1 = Bondy |first1 = John Adrian |author-link1 = John Adrian Bondy |last2 = Murty |first2 = U. S. R. |author-link2 = U. S. R. Murty |title = Graph Theory with Applications |year = 1976 |publisher = North-Holland |isbn = 0-444-19451-7 |url = https://archive.org/details/graphtheorywitha0000bond |url-status = dead |url-access = registration |access-date = 2019-12-18 |archive-url = https://web.archive.org/web/20100413104345/http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html |archive-date = 2010-04-13
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  • {{cite web | last=West | first=Douglas B. | url=http://www.math.uiuc.edu/~west/openp/1fact.html | title=1-Factorization Conjecture (1985?) | work=Open Problems – Graph Theory and Combinatorics | access-date=2010-01-09 | ref=West1FC

References

  1. {{harvtxt. Harary. 1969, Theorem 9.2, p. 85. {{harvtxt. Diestel. 2005, Corollary 2.1.3, p. 37.
  2. {{harvtxt. Harary. 1969, Theorem 9.1, p. 85.
  3. {{harvtxt. Chetwynd. Hilton. 1985. {{harvtxt. Niessen. 1994. {{harvtxt. Perkovic. Reed. 1997. [[#West1FC. West]].
  4. {{harvtxt. Petersen. 1891, §9, p. 200. {{harvtxt. Harary. 1969, Theorem 9.9, p. 90. See {{harvtxt. Diestel. 2005, Corollary 2.1.5, p. 39 for a proof.
  5. {{harvtxt. Petersen. 1891, §6, p. 198.

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