Almost-regular factorization of graphs

For integers a and b, 0 ? a ? a ? b, an a, b]-graph G satisties a ? deg(x, G) ? b for every vertex x of G, and an a, b]-factor is a spanning subgraph F such that a ? deg(x, F) ? b for every vertex x of F. An a, b]-factor is almost-regular if b = a + 1. A graph is a, b]-factorable if its edges can be decomposed into a, b]-factors. When both K and t are positive integers and s is a nonnegative integer, we prove that every (12K + 2)t + 2ks, (12k + 4)t + 2ks]-graph is 2k,2k + 1]-factorable. As its corollary, we prove that every r.r + 1]-graph with r ? 12k² + 2k is 2k + 1]-factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.