# Matchings

### From G-designs

Relevant articles:
_{[1]},
_{[2]},
_{[3]},
_{[4]}.

# Matchings

The graphs considered in this section are * -matchings*. A -matching is a graph consisting of vertex disjoint edges and is denoted by . The spectrum problem has been completely settled for matchings, see Theorem 1 below.

## Spectrum Results

Theorem 1 is a consequence of results in _{[1]}, _{[2]}, _{[3]} or _{[4]}.

**Theorem 1**

*Let . There exists an -design of order if and only if*

*or ; and**.*

## References

- ↑
^{1.0}^{1.1}Alon, N.*A note on the decomposition of graphs into isomorphic matchings*, Acta Math. Hungar.**42**, 221–223 (1983). - ↑
^{2.0}^{2.1}de Werra, D.*On some combinatorial problems arising in scheduling*, CORS J.**8**, 165–175 (1970). - ↑
^{3.0}^{3.1}Ellingham, M. N. and Wormald, N. C.*Isomorphic factorization of regular graphs and 3-regular multigraphs*, J. London Math. Soc. (2),**37**, 14–24 (1988). - ↑
^{4.0}^{4.1}McDiarmid, C. J. H.*The solution of a timetabling problem*, J. Inst. Math. Appl.**9**, 23–34 (1972).