[a,b]-factorization of a graph

Abstract Let a and b be integers such that 0 ⩽ a ⩽ b . Then a graph G is called an [ a , b ]‐graph if a ⩽ d G (x) ϵ b for every x ϵ V (G), and an [a, b]‐factor of a graph is defined to be its spanning subgraph F such that a ⩽ d F (x) ⩽ b for every vertex x, where d G (x) and d F (x) denote the degrees of x in G and F , respectively. If the edges of a graph can be decomposed into [ a.b ]‐factors then we say that the graph is [2 a , 2a ]‐factorable. We prove the following two theorems: (i) a graph G is [2 a , 2b )‐factorable if and only if G is a [2 am,2bm ]‐graph for some integer m , and (ii) every [8 m + 2k, 10 m + 2 k ]‐graph is [1,2]‐factorable.