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Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products

Ilya I. Bogdanov, Fedor Petrov, Anton Sadovnichiy, Fedor Ushakov

Abstract

In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3); Conjecture 3 implies Conjecture 1 and seems simple itself. However, both conjectures remain unproven in the general case. In the present paper, some special cases are proved, Conjecture 4 is formulated in the same spirit, and some of its special cases are proved. In addition, Conjecture 1 is reduced to a matrix inequality (Conjecture 5); this inequality is, in turn, also proved in a special case.

Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products

Abstract

In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3); Conjecture 3 implies Conjecture 1 and seems simple itself. However, both conjectures remain unproven in the general case. In the present paper, some special cases are proved, Conjecture 4 is formulated in the same spirit, and some of its special cases are proved. In addition, Conjecture 1 is reduced to a matrix inequality (Conjecture 5); this inequality is, in turn, also proved in a special case.
Paper Structure (10 sections, 17 theorems, 7 equations)

This paper contains 10 sections, 17 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of the main results
  4. Proof of Theorem \ref{['Weak4PartsT']}
  5. Results related to Conjecture \ref{['BallsC']}
  6. Results related to Conjecture \ref{['4PartsC']}
  7. Proof of Theorem \ref{['InequalityImpliesTutte']}
  8. Several proofs of Theorem \ref{['SquareMatrixT']}
  9. Further discussion
  10. Acknowledgements

Key Result

Theorem 1

Let $n_1, k_1, n_2, k_2$ be positive integers with $n_1 k_2 = n_2 k_1$. Let $M_1$ be an $n_1 \times k_1$ matrix with all the row sums equal to $k_2$, and all the column sums equal to $n_2$; let $M_2$ be an $n_2 \times k_2$ matrix with all the row sums equal to $k_1$, and all the column sums equal to

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1
  • Conjecture 2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • ...and 38 more