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arXiv · 2603.18253

Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products

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

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Ilya I. Bogdanov, Fedor Petrov, Anton Sadovnichiy, Fedor Ushakov. 2026-03-18. Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products. https://arxiv.org/abs/2603.18253

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