Cycle systems in the complete bipartite graph minus a one-factor

AbstractLet Kn,n−I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,n−I is a collection of m-cycles whose edges partition Kn,n−I. Necessary conditions for the existence of such an m-cycle system are that m⩾4 is even, n⩾3 is odd, m⩽2n, and m|n(n−1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m≡0(mod4) with n<m<2n

On 1-factorizations of Bipartite Kneser Graphs

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.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a odd prime powe