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 10, 11, such that the f-fractional number of G is the supremum of Sigma(e is an element of E(G))h(e) over all fractional functions h satisfying Sigma(e similar to v) h(e) <= f (v) for every vertex v is an element of V(G). An f-fractional factor is a spanning subgraph such that Sigma(v similar to e) h(e) = f (v) for every vertex v. In this work, we provide a new formula for computing the fractional numbers by using Lovasz'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. (C) 2010 Elsevier Ltd. All rights reserved.