Monotone Coupling and the Ising Model 1. Perfect Matching
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چکیده
Case 1: Choose any x ∈ VI and any y ∈ ∂{x} (by hypothesis, ∂{x} has at least two elements). Let G∗ be the bipartite graph with input set V ∗ I = VI − {x}, output set V ∗ O = VO − {y}, and whose edges are the same as those of G, but with edges incident to either x or y deleted. The bipartite graph G∗ satisfies the hypothesis (1.1), because in Case 1 every proper subsetA ⊂ VI has |∂A| ≥ |A|+1, so deleting the single vertex y from ∂A still leaves at least |A| vertices. By the induction hypothesis, there is a perfect matching in G∗; this perfect matching extends to a perfect matching in the original graphG by setting f(x) = y.
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