Abstract :
Let k be an odd integer ⩾ 3, and G be a connected graph of odd order n with n ⩾ 4k − 3, and minimum degree at least k. In this paper it is proved that if for each pair of nonadjacent vertices u, v in Gmax{dG(u), dG(v)}⩾n/2, then G has an almost k±-factor F± and a matching M such that F− and M are edge-disjoint and F− + M is a connected [k, k + 1]-factor of G (an almost k±-factor F± is a factor that every vertex has degree k except at most one with degree k ± 1).
As an immediate consequence, the result gives a solution to a problem of Kano on the existence of connected [k, k + 1]-factors
