Almost-regular factorization of graphs

For integers a and b, 0 ≤ a ≤ b, an [a, b]-graph G satisfies 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, 2k + 1]-factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.