Parity results on connected ƒ-factors

Abstract

Let G be a connected graph with vertex set V and let d( ν) denote the degree of a vertex νϵV. For ƒ a mapping from V to the positive integers, an ƒ-factor is a spanning subgraph having degree ƒ(ν) at vertex ν. In this paper we extend the parity results of Thomason [2] on Hamiltonian circuits to connected ƒ-factors. (A Hamiltonian circuit is a connected 2-factor.) We show that if ƒ(ν) and d( ν) have opposite parity for all νϵV then for any given subgraph C there is an even number of connected ƒ-factors having C as a cotree. Let ƒ 1 and ƒ 2 be any mappings from V to the positive integers that partition d, i.e., d(ν) = ƒ 1(ν) +ƒ 2(ν) for all νϵV. Let C 1 and C 2 be any pair of edge disjoint subgraphs. We also show in this paper that the number of decompositions of G into a connected ƒ 1- factor having C 1 as a cotree and a connected ƒ 2- factor having C 2 as a cotree is even.