General fractional -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 ∼ v h ( e ) ≤ f ( v ) for every vertex v ∈ V ( G ) . An f -fractional factor is a spanning subgraph such that ∑ v ∼ e h ( 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.