Graham–Pollak theorem
E748752
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
Statements (35)
| Predicate | Object |
|---|---|
| instanceOf |
result in graph theory
ⓘ
theorem ⓘ |
| appliesTo |
complete bipartite subgraphs
ⓘ
complete graphs ⓘ edge partitions ⓘ |
| assumes |
complete graph on n vertices
ⓘ
simple undirected graphs ⓘ |
| conclusion | Any partition of the edge set of K_n into complete bipartite graphs uses at least n−1 parts. ⓘ |
| equalityCase | There exists a partition of the edges of K_n into exactly n−1 complete bipartite subgraphs. ⓘ |
| field | graph theory ⓘ |
| hasProofMethod |
incidence matrix techniques
ⓘ
linear algebra ⓘ matrix rank arguments ⓘ |
| implies | The bipartite dimension of the complete graph K_n equals n−1. ⓘ |
| invariant | bipartite dimension of K_n equals n−1 regardless of the partition chosen. ⓘ |
| lowerBound | n−1 complete bipartite subgraphs are necessary to partition the edges of K_n. ⓘ |
| namedAfter |
Henry O. Pollak
NERFINISHED
ⓘ
Ronald Graham NERFINISHED ⓘ |
| originalAuthors |
Henry O. Pollak
NERFINISHED
ⓘ
Ronald L. Graham NERFINISHED ⓘ |
| publishedIn | Bell System Technical Journal NERFINISHED ⓘ |
| relatedTo |
bipartite edge cover
ⓘ
rank of adjacency matrices ⓘ |
| relatesConcept |
K_n
ⓘ
bipartite dimension of a graph ⓘ complete bipartite graph ⓘ edge partition ⓘ |
| statement | The edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs. ⓘ |
| tightness | The bound n−1 is best possible. ⓘ |
| topic |
edge decompositions into bipartite graphs
ⓘ
extremal graph theory ⓘ graph decompositions ⓘ |
| usedFor |
computing bipartite dimension of complete graphs
ⓘ
lower bounds in communication complexity (via related ideas) ⓘ |
| yearProved | 1971 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.