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 ⩾ 12k 2 + 2k is [2k + 1]‐factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.