An algorithmic proof of Tutte's f-factor theorem

Abstract

Let G be a multigraph on n vertices, possibly with loops. An f-factor is a subgraph of G with degree f i at the ith vertex for i = 1, 2,…, n. Tutte's f-factor theorem is proved by providing an algorithm that either finds an f-factor or shows that it does not exist and does this in O( n 3) operations. Note that the complexity bound is independent of the number of edges of G and the degrees f i . The algorithm is easily altered to handle the problem of looking for a symmetric integral matrix with given row and column sums by assigning loops degree one. A (g,f)-factor is a subgraph of G with degree d i at the ith vertex, where g i ⩽ d i ⩽ f i , for i = 1,2,…, n. Lovasz's (g,f)-factor theorem is proved by providing an O( n 3) algorithm to either find a (g,f)-factor or show one does not exist.