Pósa’s theorem in graph theory
E895558
Pósa’s theorem in graph theory is a result that gives a sufficient degree condition for a finite graph to contain a Hamiltonian cycle.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Pósa’s theorem in graph theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10944627 — 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: Pósa’s theorem in graph theory Context triple: [Lajos Pósa, knownFor, Pósa’s theorem in graph theory]
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A.
Erdős on Graphs: His Legacy
Erdős on Graphs: His Legacy is a mathematical monograph by Fan Chung and Ronald Graham that surveys and extends Paul Erdős’s influential work in graph theory and combinatorics.
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B.
Turán's theorem
Turán's theorem is a fundamental result in extremal graph theory that determines the maximum number of edges a graph can have without containing a complete subgraph of a given size.
-
C.
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.
-
D.
Menger theorem in graph theory
Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
-
E.
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.
- 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: Pósa’s theorem in graph theory Target entity description: Pósa’s theorem in graph theory is a result that gives a sufficient degree condition for a finite graph to contain a Hamiltonian cycle.
-
A.
Erdős on Graphs: His Legacy
Erdős on Graphs: His Legacy is a mathematical monograph by Fan Chung and Ronald Graham that surveys and extends Paul Erdős’s influential work in graph theory and combinatorics.
-
B.
Turán's theorem
Turán's theorem is a fundamental result in extremal graph theory that determines the maximum number of edges a graph can have without containing a complete subgraph of a given size.
-
C.
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.
-
D.
Menger theorem in graph theory
Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
-
E.
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.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
result in extremal graph theory
ⓘ
theorem in graph theory ⓘ |
| appearsIn |
Bondy and Murty’s Graph Theory with Applications
ⓘ
standard graduate texts on graph theory ⓘ |
| appliesTo | finite simple graphs ⓘ |
| assumes | graph has at least three vertices ⓘ |
| category |
Hamiltonian graph theorems
ⓘ
degree sequence theorems in graph theory ⓘ |
| concerns | Hamiltonian cycle existence ⓘ |
| field |
discrete mathematics
ⓘ
extremal graph theory ⓘ graph theory ⓘ |
| generalizes | Dirac’s degree condition for Hamiltonicity ⓘ |
| gives | sufficient condition for existence of Hamiltonian cycle ⓘ |
| hasConsequence |
gives families of Hamiltonian graphs from degree sequences
ⓘ
provides extremal bounds for non-Hamiltonian graphs ⓘ |
| hasFormulation | If G is a graph on n ≥ 3 vertices with degree sequence d1 ≤ d2 ≤ … ≤ dn and for every integer k with 1 ≤ k < n/2, dk ≥ k+1 or d_{n−k} ≥ n−k, then G is Hamiltonian ⓘ |
| hasWeakerFormulation | If G is a graph on n ≥ 3 vertices with degree sequence d1 ≤ d2 ≤ … ≤ dn and for every integer k with 1 ≤ k < n/2, dk ≥ k+1 or dn−k ≥ n−k, then G has a Hamiltonian cycle ⓘ |
| implies | graph is Hamiltonian under its degree conditions ⓘ |
| isToolFor |
proving Hamiltonicity of dense graphs
ⓘ
studying degree sequence characterizations of Hamiltonian graphs ⓘ |
| namedAfter | Lajos Pósa ⓘ |
| namedEntity | true ⓘ |
| originalLanguage | Hungarian ⓘ |
| publishedIn |
Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei
ⓘ
linked to:
Alfréd Rényi Institute of Mathematics
|
| relatedConcept |
Chvátal–Erdős theorem
ⓘ
linked to:
Bondy–Chvátal theorem
Hamiltonian path ⓘ closure of a graph ⓘ |
| relatedTo |
Chvátal’s theorem
ⓘ
linked to:
Bondy–Chvátal theorem
Ore’s theorem ⓘ
linked to:
Ore's theorem
|
| strengthens |
Dirac’s theorem
ⓘ
linked to:
Dirac's theorem
|
| subject |
Hamiltonian cycles
ⓘ
degree conditions in graphs ⓘ |
| typeOfCondition | degree sequence condition ⓘ |
| usedIn |
graph theory textbooks
ⓘ
research on Hamiltonian properties of graphs ⓘ sufficient conditions for Hamiltonian cycles ⓘ |
| usesConcept |
Hamiltonian graph
ⓘ
nondecreasing degree sequence ⓘ vertex degree ⓘ |
| yearProved | 1962 ⓘ |
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: Pósa’s theorem in graph theory Description of subject: Pósa’s theorem in graph theory is a result that gives a sufficient degree condition for a finite graph to contain a Hamiltonian cycle.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.