Erdős–Gallai
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| Erdős–Gallai | |
|---|---|
| Name | Paul Erdős and Tibor Gallai theorem |
| Known for | Erdős–Gallai theorem |
Erdős–Gallai
Introduction
The Erdős–Gallai result connects work by Paul Erdős, Tibor Gallai, Vera T. Sós, Alfréd Rényi, and contemporaries such as Pál Turán, Donald Knuth, László Lovász, Endre Szemerédi in graph theory, combinatorics, and discrete mathematics. It appears in two closely related formulations: one on degree sequences influenced by classical results of Olivier Ore and Nicolas Bourbaki-era combinatorialists, and another in extremal graph theory linked to problems studied by Erdős with collaborators including Tibor Gallai, Miklós Simonovits, Zoltán Füredi, and Richard Rado. The theorem plays a role alongside landmarks such as Turán's theorem, Mantel's theorem, Ramsey theory, and developments by Paul Turán supporters and critics.
Erdős–Gallai theorem (graphical sequences)
The graphical-sequence theorem characterizes degree sequences, building on earlier criteria by Olivier Ore and work by Harary and Fulkerson. It provides necessary and sufficient inequalities for a nonincreasing integer sequence to be realizable as the degree sequence of a simple undirected graph; the criterion relates cumulative degree sums to combinatorial bounds inspired by techniques used by Pál Turán and refined in the literature by Vera T. Sós and Lajos Pósa. This formulation is commonly taught alongside the Havel–Hakimi algorithm studied by William Havel and S. L. Hakimi and is cited in expositions by Béla Bollobás, Richard Stanley, Ronald Graham, Fan Chung, and J. H. van Lint. The sequence version is often applied in problems first posed by Erdős and contemporaries such as Paul Turán and revisited by researchers including Miklós Simonovits, Zoltán Füredi, Endre Szemerédi, László Lovász, and Robin Wilson.
Erdős–Gallai theorem (extremal graph theory)
The extremal statement provides numeric bounds on edges avoiding long paths or specific subgraphs, extending themes from Turán's theorem and Mantel's theorem and complementing results by Dirac and Ore. Erdős and Gallai used extremal techniques that later connected to the Szemerédi regularity lemma developed by Endre Szemerédi and applications by János Komlós and Miklós Simonovits. This variant informs studies by Béla Bollobás, Paul Erdős coauthors such as Arthur H. Stone, István Juhász, and later refinements by Zoltán Füredi and Andrásfai, Erdős, Sós-style theorems involving Miklós Simonovits and László Lovász.
Proofs and techniques
Proofs of the graphical-sequence criterion use constructive algorithms akin to the Havel–Hakimi algorithm and inductive combinatorial arguments influenced by methods of Pál Turán, Erdős, and Tibor Gallai. Alternative proofs employ network-flow approaches related to work by Lloyd Shapley and John von Neumann ideas appearing in combinatorial optimization literature developed by Jack Edmonds and Dantzig-style inequalities. Extremal proofs invoke counting arguments, double-counting strategies used by Paul Erdős and Alfréd Rényi, and structural decomposition techniques that echo the Szemerédi regularity lemma and matching theory of László Lovász and Endre Szemerédi.
Applications and consequences
The graphical-sequence theorem is applied in constructive design problems studied by Frank Harary, Béla Bollobás, Fan Chung, and Richard Stanley, including network realization tasks that relate to work by Claude Shannon in information networks and combinatorial designs examined by Ronald Graham and Donald Knuth. Extremal formulations inform path-length bounds and sparse graph constructions investigated by Paul Erdős, Miklós Simonovits, Zoltán Füredi, Béla Bollobás, and computational implications relevant to algorithmic graph theory work by Noga Alon, László Lovász, Sanjeev Arora, and David Karger. The results have been used in studies of degree-preserving randomizations in research by Persi Diaconis, Susan Holmes, and Mark Newman and in structural graph theory connected to Paul Erdős collaborations with Joel Spencer and Ronald Graham.
Variants and generalizations
Generalizations of the graphical criterion include directed and bipartite analogues developed in tandem with results by Havel–Hakimi successors and explored by Fulkerson, Ryser, Gale and others such as John W. Moon, Harary, Harold S. Shapiro, and S. L. Hakimi. Extensions intersect with degree-sequence research by Béla Bollobás, Vera T. Sós, Miklós Simonovits, Zoltán Füredi, and probabilistic methods advanced by Paul Erdős collaborators including Alfréd Rényi, Noga Alon, and Joel Spencer. Extremal variants link to stability results by Miklós Simonovits, inducibility problems studied by Béla Bollobás and Paul Erdős, and to algorithmic generalizations considered by Richard Karp and Jack Edmonds.