[a,b]‐factorizations of graphs

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Abstract
Let a and b be integers with b ⩾ a ⩾ 0. A graph G is called an [a,b]‐graph if a ⩽ dG(v) ⩽ b for each vertex v ∈ V(G), and an [a,b]‐factor of a graph G is a spanning [a,b]‐subgraph of G. A graph is [a,b]‐factorable if its edges can be decomposed into [a,b]‐factors. The purpose of this paper is to prove the following three theorems: (i) if 1 ⩽ b ⩽ 2a, every [(12a + 2)m + 2an,(12b + 4)m + 2bn]‐graph is [2a, 2b + 1]‐factorable; (ii) if b ⩽ 2a −1, every [(12a −4)m + 2an, (12b −2)m + 2bn]‐graph is [2a −1,2b]‐factorable; and (iii) if b ⩽ 2a −1, every [(6a −2)m + 2an, (6b + 2)m + 2bn]‐graph is [2a −1,2b + 1]‐factorable, where m and n are nonnegative integers. They generalize some [a,b]‐factorization results of Akiyama and Kano [3], Kano [6], and Era [5].

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