Erdős–Sós

Erdős–Sós is a central conjecture in Graph theory and Extremal graph theory proposed by Paul Erdős and Vera T. Sós in 1963, asserting that any n-vertex graph whose average degree exceeds k−1 contains every tree with k edges. The conjecture links degree conditions in finite graphs to embedding of acyclic structures such as spanning trees and specific trees like path graphs and star graphs, and it has driven developments involving methods from Probabilistic method, Matching theory, and Regularity lemma techniques. Over decades, contributions by researchers including Béla Bollobás, Vladimir Nikiforov, Andrásfai, József Komlós, Endre Szemerédi, Noga Alon, Radoslav Kirov, Alex Scott, David Wood, Michael Krivelevich, Miklós Simonovits, Carsten Thomassen, Zoltán Füredi, and Peter Allen have resolved many special cases and introduced influential tools.

Statement and history

The original formulation by Paul Erdős and Vera T. Sós posits that for every positive integer k, any n-vertex simple graph with more than (k−1)n/2 edges contains every tree of k edges. Early progress involved extremal examples studied by Paul Turán and motivations from the Erdős–Gallai theorem on path lengths, while counterexamples and tightness conditions were informed by constructions related to Turán graphs and balanced complete multipartite graphs. Partial confirmations for small k were obtained through direct combinatorial arguments by Béla Bollobás and extensions by Zoltán Füredi; asymptotic proofs used innovations from Komlós, Sárközy, Szemerédi's embedding results and the Szemerédi regularity lemma. Subsequent historical milestones include results by Ajtai, Komlós, Simonovits on embedding bounded-degree trees, by Richard Rado-style matching techniques, and refinement from Erdős–Gallai type extremal path analyses.

Proofs and partial results

Multiple lines of attack produced proofs for special families: the conjecture is known for trees with bounded maximum degree via methods of Komlós, Sárközy, Szemerédi, for path-like trees using adaptations of the Erdős–Gallai theorem and results by Faudree and Schelp, and for large k relative to n using probabilistic embedding from Noga Alon and Michael Krivelevich. The conjecture holds for stars and double stars by trivial degree-counting, for caterpillars through constructive greedy embedding influenced by Béla Bollobás techniques, and for spiders via matching methods related to Hall's marriage theorem. Results proving the conjecture for dense graphs employed the Szemerédi regularity lemma and the Blow-up lemma developed by Komlós, Sárközy, Szemerédi, while spectral approaches by Vladimir Nikiforov and Bing Huang produced eigenvalue conditions guaranteeing tree embeddings. Extremal-degree threshold improvements arose from stability methods introduced by Elliott H. Lieb-style combinatorial optimization and by work of Andrásfai and Erdős–Sós collaborators.

Related conjectures and generalizations

The Erdős–Sós conjecture intersects with the Loebl–Komlós–Sós conjecture, which predicts subtree embeddings given degree frequency conditions, and with the Erdős–Gallai theorem which characterizes integer sequences realizable by simple graphs; both motivated generalized degree conditions such as the Pósa–Seymour conjecture for powers of Hamiltonian cycles and the Tree Packing Conjecture about edge-disjoint trees by Gyárfás and Lehel. Variants include directed versions inspired by Redei and Camion, hypergraph analogues related to work by Turán and Keevash, and robust expansion formulations linked to Krivelevich–Sudakov expansion frameworks. Strengthened statements consider minimum degree conditions akin to Dirac-type theorems pioneered by G. A. Dirac and by Ore; fractional relaxations relate to fractional matching theory and to conjectures of Lovász and Alon–Yuster on embedding bounded-degree subgraphs.

Extremal examples and constructions

Known near-extremal constructions derive from complete bipartite and multipartite graphs such as Turán graph variants, balanced complete bipartite graphs forming tight examples for star-free embeddings, and disjoint union constructions modeled on Erdős extremal set systems. Specific counterexamples for weaker bounds exploit constructions used in the Erdős–Rényi model as probabilistic obstructions or use blow-up techniques from T. Sós-style combinatorial designs. Sharpness analyses reference classical constructions by Mantel and Turán, stability constructions by Simonovits, and localized obstruction families studied by Bollobás and Füredi; these inform which trees are hardest to embed, often those concentrating edges near high-degree vertices.

Applications and connections to other areas

The conjecture and its partial resolutions influence algorithms for subgraph isomorphism studied in Richard M. Karp-inspired complexity theory, parameterized algorithms in the style of Downey and Fellows, and approximation algorithms drawing on Noga Alon's probabilistic method. Connections to design theory and Combinatorial designs appear via embedding trees in block designs and via Steiner system constraints, while graph-sparsification and routing in networks relate to embeddings studied by Jon Kleinberg and Éva Tardos. Further ties include spectral graph theory results by Fan R. K. Chung and László Lovász, random graph thresholds from Erdős–Rényi theory, and applications to phylogenetic tree reconstruction in computational biology influenced by methods from Sankar Raman, Mike Steel, and László Székely. The problem continues to inspire cross-disciplinary tools from probabilistic combinatorics, extremal set theory, and algorithmic graph theory.

Category:Conjectures in graph theory