# On 1-factorizations of Bipartite Kneser Graphs

## TL;DR

This paper introduces a new framework for constructing 1-factorizations of bipartite Kneser graphs, successfully solving a complex case for prime power parameters and analyzing classic factorizations with efficient algorithms.

## Contribution

It proposes a novel framework for 1-factorizations, solves a specific case for prime powers, and simplifies and analyzes classic factorizations with an optimal algorithm.

## Key findings

- Solved a nontrivial case with t=2 and v as an odd prime power.
- Provided simplified definitions and structural analysis of classic factorizations.
- Designed an optimal algorithm for lexical factorizations.

## Abstract

It is a challenging open problem to construct an explicit 1-factorization of the bipartite Kneser graph $H(v,t)$, which contains as vertices all $t$-element and $(v-t)$-element subsets of $[v]:=\{1,\ldots,v\}$ and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where $t=2$ and $v$ is an odd prime power. We also revisit two classic constructions for the case $v=2t+1$ --- the \emph{lexical factorization} and \emph{modular factorization}. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations. (An analogous algorithm for the modular factorization is trivial.)