On 1-Factorizations of Bipartite Kneser Graphs

Abstract

It is a challenging open problem to construct an explicit 1factorization of the bipartite Kneser graph H(v, t), which contains as vertices all t-element and (v − t)-element subsets of [v] := {1, . . . , 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 lexical factorization and 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.)On 1-factorizations of Bipartite Kneser Graphs 3 PreliminariesThe subset lattice is the family of all subsets of [v], partially ordered by inclusion. Let P t denote the t-th layer of this subset lattice, whose members are the telement subsets of [v]. Throughout the paper, denote d = v − 2t. Let the words clockwise and counterclockwise be abbreviated as CW and CCW respectively. A representation of the edges of H(v, t). We identify each edge (A, A ) of H(v, t) by a permutation ρ of t 's, t 's, and d ×'s: the (positions of) t ' 's indicate the t elements in A; the t ' 's indicate the t elements that are not in A (recall that A has v − t elements); and the '×'s indicate those in A − A. We do not distinguish the edges with their corresponding permutations.Denote [t , t , d×] as the multiset of 2t + d characters with t ' 's, t ' 's, and d '×'s. Giving a 1-factorization of H(v, t) is equivalent to giving a labeling function f from the 2t+d t,t,d permutations of [t , t , d×] to 1, . . . , t+d d so that(a) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's; and (b) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's.

A local injective proof of log-concavity for increasing spanning forests

“…First construct all the bijections Φ 2k+1,k , for all k. This amounts to chosing a perfect matching in the bipartite Kneser graph H(2k + 1, k), i.e., the middle two levels of the Boolean lattice of [2k + 1]. There are many matchings to choose from (see [16] and the more recent [15]) and their study was motivated by the middle levels conjecture (see [10] and references therein). Then Φ n,k can be constructed recursively, or by induction on n as follows.…”