Four cycle decomposition of λK(n,2)

Abstract

The Kneser graph $K(n,k)$ is the graph with the $k$-subsets of a fixed $n$-set as its vertices, with two $k$-subsets adjacent if they are disjoint. Given $n$ and $k$, the existence of Hamilton cycle in $K(n,k)$ was considered by many authors since 1978. Recently, Merino et al. (2023) proved that all connected Kneser graphs $K(n,k)$ are Hamiltonian. In this paper, we examine the necessary and sufficient conditions for the existence of a 4-cycle decomposition of $\lambda$-fold Kneser graphs $\lambda K(n,2)$ and $\lambda$-fold Bipartite Kneser graphs $\lambda BK(n,2)$. The main result of this paper may shed some lights on the $k$-cycle decomposition problem of Kneser graphs.