k-factor-covered regular graphs

A graph G is called f-factor-covered if every edge of G is contained in some f-factor. G is called f-factor-deleted if G - e contains a f-factor for every edge e. Babler proved that every r-regular, (r - 1)-edge-connected graph of even order has a 1-factor. In the present article, we prove that every 2r-regular graph of odd order is both 2m-factor-covered and 2m-factor-deleted for all integers m, 1 <= m <= r - 1, and every r-regular, (r - 1)-edge-connected graph of even order is both m-factor-covered and m-factor-deleted for all integers m, 1 <= m <= [r/2].