Erdős–Gallai theorem
E554299
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Erdős–Gallai theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896706 — 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: Erdős–Gallai theorem Context triple: [Pál Erdős, knownFor, Erdős–Gallai theorem]
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A.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
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B.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
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C.
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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D.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
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E.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
- 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: Erdős–Gallai theorem Target entity description: 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.
-
A.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
-
B.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
C.
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.
-
D.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
-
E.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
graph theory theorem
ⓘ
mathematical theorem ⓘ |
| appliesTo | simple undirected graphs ⓘ |
| assumes | sequence of nonnegative integers in nonincreasing order ⓘ |
| characterizes | degree sequences of simple graphs ⓘ |
| classification | characterization theorem ⓘ |
| concerns | existence of a simple graph with given degrees ⓘ |
| conditionType | inequality conditions ⓘ |
| countryOfOrigin | Hungary NERFINISHED ⓘ |
| dealsWith |
finite simple graphs
ⓘ
nonnegative integer sequences ⓘ |
| field | graph theory ⓘ |
| givesNecessaryAndSufficientConditionFor | a sequence to be graphical ⓘ |
| hasApplication |
design of networks with specified degree distributions
ⓘ
verification of empirical degree sequences in real-world networks ⓘ |
| hasConsequence | characterization of all graphical sequences ⓘ |
| hasGeneralization |
results on degree sequences of directed graphs
ⓘ
results on degree sequences of hypergraphs ⓘ |
| hasProofTechnique |
combinatorial arguments
ⓘ
induction on number of vertices ⓘ |
| implies | parity condition on degree sums ⓘ |
| importance | fundamental result in graph theory ⓘ |
| inequalityForm | for all k, sum_{i=1}^k d_i ≤ k(k−1) + sum_{i=k+1}^n min(d_i,k) ⓘ |
| involvesConcept |
degree of a vertex
ⓘ
graphical sequence ⓘ nonincreasing sequence ⓘ simple graph ⓘ |
| languageOfOriginalPublication | Hungarian ⓘ |
| mathematicalDomain |
combinatorics
ⓘ
discrete mathematics ⓘ |
| namedAfter |
Paul Erdős
NERFINISHED
ⓘ
Tibor Gallai NERFINISHED ⓘ |
| relatedTo |
Havel–Hakimi algorithm
NERFINISHED
ⓘ
Turán-type extremal problems ⓘ graph realization algorithms ⓘ |
| requires |
Erdős–Gallai inequalities to hold for all k
ⓘ
even sum of degrees ⓘ |
| statesThat | a nonincreasing sequence d1,…,dn of nonnegative integers is graphical if and only if the sum of the di is even and certain inequalities hold for all k between 1 and n ⓘ |
| timePeriod | 20th century ⓘ |
| topic |
degree sequence characterization
ⓘ
graph realization problem ⓘ graphical degree sequences ⓘ |
| usedFor |
constructing simple graphs with given degree sequence
ⓘ
testing whether a sequence is graphical ⓘ |
| usedIn |
degree-based graph models
ⓘ
network theory ⓘ random graph generation with prescribed degrees ⓘ social network analysis ⓘ |
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: Erdős–Gallai theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.