Sufficient Conditions for a Graph k G Admitting all 1 , k -factors

Let g, f: V (G) →0, 1, 2, 3, ⋯ be two functions satisfying g (x) ≤f (x) for every x∈V (G). A (g, f) -factor of G is defined as a spanning subgraph F of G such that g (x) ≤dF (x) ≤f (x) for every x∈V (G). An (f, f) -factor is simply called an f-factor. Let φ be a nonnegative integer-valued function defined on V (G). Set Deveng, f=φ: g (x) ≤φ (x) ≤f (x) for every x∈V (G) and ∑x∈V (G) φ (x) is even. If for each φ∈Deveng, f, G admits a φ-factor, then we say that G admits all (g, f) -factors. All (g, f) -factors are said to be all 1, k-factors if g (x) ≡1 and f (x) ≡k for any x∈V (G). In this paper, we verify that for a connected multigraph G satisfying NG (X) =V (G) or |NG (X) |> (1+1k+1) |X|−1 for every X⊂V (G), kG admits all 1, k-factors, where k≥2 is an integer and kG denotes the graph derived from G by replacing every edge of G with k parallel edges.