General fractional f-factor numbers of graphs

Let G be a graph and f an integer-valued function on V(G). Let h be a function that assigns each edge to a number in [0,1], such that the f-fractional number of G is the supremum of ∑e∈E(G)h(e) over all fractional functions h satisfying ∑e∼vh(e)≤f(v) for every vertex v∈V(G). An f-fractional factor is a spanning subgraph such that ∑v∼eh(e)=f(v) for every vertex v. In this work, we provide a new formula for computing the fractional numbers by using Lovász's Structure Theorem. This formula generalizes the formula given in [Y. Liu, G.Z. Liu, The fractional matching numbers of graphs, Networks 40 (2002) 228-231] for the fractional matching numbers.