Extremal properties of (1, f)-odd factors in graphs

Let G be a simple graph and f : V (G) 7→ {1, 3, 5, ...} an odd integer valued function defined on V (G). A spanning subgraph F of G is called a (1, f)odd factor if dF (v) ∈ {1, 3, ..., f(v)} for all v ∈ V (G), where dF (v) is the degree of v in F . For an odd integer k, if f(v) = k for all v, then a (1, f)odd factor is called a [1, k]-odd factor. In this paper, the structure and properties of a graph with a unique (1, f)-odd factor is investigated, and the maximum number of edges in a graph of the given order which has a unique [1, k]-odd factor is determined.