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Perfect 1-factorizations

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Published/Copyright: May 21, 2019
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 3

Abstract

Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1.

One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.

A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.

It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

MSC 2010: Primary 05C70; 05B30
Keywords: 1-factorization; perfect; uniform

  1. (Communicated by Peter Horák)

Acknowledgement

Thanks to Peter Horák, Mariusz Meszka and Ian Wanless for many valuable comments, and to the referees for helpful comments and suggestions.

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Received: 2018-05-10
Accepted: 2018-09-19
Published Online: 2019-05-21
Published in Print: 2019-06-26

© 2019 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. The life jubilee of Prof. RNDr. Sylvia Pulmannová, DrSc.
  3. Perfect 1-factorizations
  4. A topological duality for strong Boolean posets
  5. On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
  6. Tribonacci numbers and primes of the form p = x2 + 11y2
  7. Basic semirings
  8. A conjecture for varieties of completely regular semigroups
  9. Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets
  10. Differential subordination results for Mittag-Leffler type functions with bounded turning property
  11. Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays
  12. Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2
  13. On the polynomial entropy for morse gradient systems
  14. Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators
  15. A note on non-linear ∗-Jordan derivations on ∗-algebras
  16. Disjoint hypercyclic weighted translations on locally compact hausdorff spaces
  17. Some new results on real hypersurfaces with generalized Tanaka-Webster connection
  18. Relative topological properties of hyperspaces
  19. Cohomology of torus manifold bundles
  20. The Menger and projective Menger properties of function spaces with the set-open topology
  21. Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications
https://doi.org/10.1515/ms-2017-0241
Keywords for this article
1-factorization; perfect; uniform

Articles in the same Issue

  1. Regular papers
  2. The life jubilee of Prof. RNDr. Sylvia Pulmannová, DrSc.
  3. Perfect 1-factorizations
  4. A topological duality for strong Boolean posets
  5. On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
  6. Tribonacci numbers and primes of the form p = x2 + 11y2
  7. Basic semirings
  8. A conjecture for varieties of completely regular semigroups
  9. Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets
  10. Differential subordination results for Mittag-Leffler type functions with bounded turning property
  11. Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays
  12. Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2
  13. On the polynomial entropy for morse gradient systems
  14. Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators
  15. A note on non-linear ∗-Jordan derivations on ∗-algebras
  16. Disjoint hypercyclic weighted translations on locally compact hausdorff spaces
  17. Some new results on real hypersurfaces with generalized Tanaka-Webster connection
  18. Relative topological properties of hyperspaces
  19. Cohomology of torus manifold bundles
  20. The Menger and projective Menger properties of function spaces with the set-open topology
  21. Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications
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