Odd subgraphs and matchings

Let G be a graph and f : V (G) → {1, 3, 5, . . .}. Then a subgraph H of G is called a (1, f)-odd subgraph if degH(x) ∈ {1, 3, . . . , f(x)} for all x ∈ V (H). If f(x) = 1 for all x ∈ V (G), then a (1, f)-odd subgraph is nothing but a matching. A (1, f)-odd subgraph H of G is said to be maximum if G has no (1, f)-odd subgraph K such that |K| > |H|. We show that (1, f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1, f)-odd subgraph, which is similar to that for the order of a maximum matching.