Abstract
For a set \({\mathcal{S}}\) of positive integers, a spanning subgraph F of a graph G is called an \({\mathcal{S}}\) -factor of G if \({\deg_F(x) \in \mathcal{S}}\) for all vertices x of G, where deg F (x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an even integer such that 2 ≤ k < r/2 or an odd integer such that r/3 ≤ k < r/2. Then every r-regular graph G has a {k, r–k}-factor. Moreover, for every edge e of G, G has a {k, r–k}-factor containing e and another {k, r–k}-factor avoiding e.
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This work was partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C).
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Akbari, S., Kano, M. {k, r – k}-Factors of r-Regular Graphs. Graphs and Combinatorics 30, 821–826 (2014). https://doi.org/10.1007/s00373-013-1324-x
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DOI: https://doi.org/10.1007/s00373-013-1324-x