Orthogonal matchings revisited

Qu, Cunquan; Wang, Guanghui; Ge, Yan

doi:10.1016/j.disc.2015.05.009

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…A greedy argument proves the conjecture when we assume |G| ≥ 4d − 3. This was improved to |G| ≥ 4d − 5 by Alspach, Heinrich and Li [7], to |G| ≥ 3.32d by Kouider and Sotteau [23], to |G| ≥ 3d − 2 by Stong [30], and finally to |G| ≥ 2 √ 2d + 4.5 by Qu, Wang, and Yan [27]. Our sampling trick improves on all these results and establishes the strong asymptotic version of Alspach's conjecture.…”

Section: Alspach's Conjecturesupporting

confidence: 58%

Short Proofs of Rainbow Matchings Results

Correia

Pokrovskiy

Sudakov

2022

International Mathematics Research Notices

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A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that “every edge coloured graph of a certain type contains a rainbow matching using every colour.” In this paper we introduce a versatile “sampling trick,” which allows us to asymptotically solve some well-known conjectures and to obtain short proofs of old results. In particular: $\bullet $ We give the first asymptotic proof of the “non-bipartite” Aharoni–Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky, and Zerbib. $\bullet $ We give a very short asymptotic proof of Grinblat’s conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat’s problem as a function of edge multiplicity of the corresponding multigraph. $\bullet $ We give the first asymptotic proof of a 30-year-old conjecture of Alspach. $\bullet $ We give a simple proof of Pokrovskiy’s asymptotic version of the Aharoni–Berger conjecture with greatly improved error term.

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Section: Alspach's Conjecturesupporting

confidence: 58%

Short Proofs of Rainbow Matchings Results

Correia

Pokrovskiy

Sudakov

2022

International Mathematics Research Notices

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show abstract

Section: Alspach's Conjecturesupporting

confidence: 57%

Short proofs of rainbow matching results

Munhá¹,

Pokrovskiy²,

Sudakov³

2021

Preprint

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A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that "every edge coloured graph of a certain type contains a rainbow matching using every colour". In this paper we introduce a versatile "sampling trick", which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures.• We give a simple proof of Pokrovskiy's asymptotic version of the Aharoni-Berger conjecture with greatly improved error term.• We give the first asymptotic proof of the "non-bipartite" Aharoni-Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky and Zerbib.• We give a very short asymptotic proof of Grinblat's conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat's problem as a function of edge multiplicity of the corresponding multigraph.• We give the first asymptotic proof of a 30 year old conjecture of Alspach.

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“…Definition 1 [9][10][11] : An n-ary operation on a set S is a function ＊ : S ×S ×•••×S → S, where the domain is the product of n factors.…”

Section: Proper Inclusions Of Sets Relations Functions and Operationsmentioning

confidence: 99%

“…Analysis: To visually represent the proper inclusions of sets, relationships, functions and operations, we draw the following Venn diagram (Fig. 1) based on references [9][10][11][12]. We determine that the function…”

Section: Proper Inclusions Of Sets Relations Functions and Operationsmentioning

confidence: 99%

See 1 more Smart Citation

A Case Study of the Diversity and Optimization of Mathematical Definitions and Symbolization

Yang

Zeng

et al. 2023

SHS Web of Conf.

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Taking the idea of “reflective and critical optimization thinking”, this article first examines “the outlook on development of big mathematics” in International Mathematics Day in 2003 from the perspective of the history of mathematics. Then according to the requirements of mathematical core literacy and mathematical beauty, a deep analysis is conducted on the limitations of the teaching methods in the Turing mathematics and statistics series “Terence Tao Analysis”. The case study shows that (1) the sets, relations, functions and operations present a concentric circle relationship of layer-by-layer proper subsets; (2) rejecting mathematical notation damages the beauty of simplicity and learning efficiency; and (3) forcing the first mapping function to be a surjection unnecessarily limits the wider range of applications that composite functions can have, and it lacks the unified beauty of mathematical definitions. Finally, itemized strategies and suggestions for improvement and optimization are proposed.

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Orthogonal matchings revisited

Cited by 4 publications

References 16 publications

Short Proofs of Rainbow Matchings Results

Short Proofs of Rainbow Matchings Results

Short proofs of rainbow matching results

A Case Study of the Diversity and Optimization of Mathematical Definitions and Symbolization

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