Extending partial systems of distinct representatives
✍ Scribed by Peter Horák
- Publisher
- Elsevier Science
- Year
- 1991
- Tongue
- English
- Weight
- 234 KB
- Volume
- 91
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
✦ Synopsis
Horiik, P., Extending partial systems of distinct representatives, Discrete Mathematics 91 (1991) 95-98.
We determine a necessary and sufficient condition for a special class of families of sets to have the property that each partial SDR of cardinahty d can be extended to a total SDR. This result is a generalization of a theorem of Brualdi and Csima concerning matching extension in a regular bipartite graphs.
📜 SIMILAR VOLUMES
Agrawal (1966 Ann. Math. Statist. 37, 525-528) explored the concept of systems of distinct representatives to show that the treatments in a binary equireplicated incomplete block design can be rearranged within blocks such that the treatments occur as close to equally often as possible in every row.
The main result of this paper can be quickly described as follows. Let G be a bipartite graph and assume that for any vertex v of G a strongly base orderable matroid is given on the set of edges adjacent with v. Call a subgraph of G a system of representatives of G if the edge neighborhood of each v
In this paper, we show that any partial extended triple system (partial totally symmetric quasigroup) of order n can be embedded in a totally symmetric quasigroup of order v, v >~4n +6, v -= 2(mod4). This bound can be lowered to 4n + 2 in most cases.
Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or listmulticolorings of complete graphs. The necessary and sufficient condition for a proper