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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Graham–Pollak theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8657127 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Graham–Pollak theorem Context triple: [Ronald L. Graham, notableIdea, Graham–Pollak theorem]
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A.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
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B.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
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C.
Birkhoff–von Neumann theorem
The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
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D.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
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E.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Graham–Pollak theorem Target entity description: 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.
-
A.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
-
B.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
-
C.
Birkhoff–von Neumann theorem
The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
-
D.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
-
E.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
- F. None of above. chosen
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
ⓘ
linked to:
Henry Pollak
Ronald Graham ⓘ
linked to:
Ronald L. Graham
|
| originalAuthors |
Henry O. Pollak
ⓘ
linked to:
Henry Pollak
Ronald L. Graham ⓘ |
| publishedIn | Bell System Technical Journal ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Graham–Pollak theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.