An alternative proof of general factor structure theorem

Let G be a graph, and H : V (G)? 2(N) a set function associated with G. A spanning subgraph F of graph G is called a general factor or an H -factor of G if d(F)(x) ? H(x) for every vertex x ? V(G). The existence of H-factors is, in general, an NP -complete problem. H-factor problems are considered as one of most general factor problem because many well-studied factors (e.g., per-fect matchings, f-factor problems and (g, f)-factor problems) are special cases of H-factors. Lovasz [The factorization of graphs (II), Acta Math. Hungar., 23 (1972), 223-246] gave a structure descrip-tion of H-optimal subgraphs and obtained a deficiency formula. In this paper, we introduce a new type of alternating path to study Lovasz's canonical structural partition of graphs and consequently obtain an alternative and shorter proof of Lovasz's deficiency for-mula for H-factors. Moreover, we also obtain new properties re-garding Lovasz's canonical structural partition of H-factors.