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.