Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products

TL;DR

This paper investigates bipartite graphs and tensor products, proving special cases of longstanding conjectures related to perfect matchings and reducing a key conjecture to a matrix inequality, some of which are also proved.

Contribution

It proves special cases of two longstanding conjectures in bipartite graph theory and reduces one conjecture to a matrix inequality, providing partial progress in the field.

Findings

Proved special cases of Higgins' and Petrov's conjectures.

Formulated and proved special cases of Conjecture 4.

Reduced Conjecture 1 to a matrix inequality and proved it in some cases.

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.