An algorithmic proof of Tutte's f-factor theorem |
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Authors: | R.P Anstee |
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Affiliation: | Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada |
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Abstract: | Let G be a multigraph on n vertices, possibly with loops. An f-factor is a subgraph of G with degree fi 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(n3) operations. Note that the complexity bound is independent of the number of edges of G and the degrees fi. 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 di at the ith vertex, where gi ? di ? fi, for i = 1,2,…, n. Lovasz's (g,f)-factor theorem is proved by providing an O(n3) algorithm to either find a (g,f)-factor or show one does not exist. |
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