# 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 a proper edge coloring with

*k*colors. A

**2-factor**is a collection of 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.
^{[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

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 *K*_{2}, *K*_{4}, *K*_{6}, *K*_{8}, ... is 1, 1, 6, 6240, 1225566720, 252282619805368320, 98758655816833727741338583040, ... (OEIS: A000438).

### 1-factorization conjecture

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*≥ 12*n*/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

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

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 open.

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