Bipartite graphs related to mutually disjoint S-permutation matrices

TL;DR

This paper explores the relationship between bipartite graphs and disjoint S-permutation matrices, providing classifications and cardinalities for specific cases in matrix disjointness problems.

Contribution

It characterizes bipartite graphs related to disjoint S-permutation matrices and calculates the number of such matrices for 4x4 and 9x9 cases.

Findings

Classified bipartite graphs with 2 and 3 vertices in each part.

Calculated the number of disjoint S-permutation matrices for 4x4 matrices.

Calculated the number of disjoint S-permutation matrices for 9x9 matrices.

Abstract

Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general $n^2 \times n^2$ case are discussed in this paper. All bipartite graphs of the type $g=<R_g \cup C_g, E_g>$, where $|R_g |=|C_g |=2$ or $|R_g |=|C_g |=3$ are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the $4 \times 4$ and $9 \times 9$ cases are calculated.