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arxiv: 1211.2374 · v1 · pith:PL4NUTGGnew · submitted 2012-11-11 · 🧮 math.CO · math.DS· math.PR

Invariant random matchings in Cayley graphs

classification 🧮 math.CO math.DSmath.PR
keywords graphmatchingadmitscayleyfiniteperfectgraphsinvariant
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We prove that any non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected d-regular vertex tran- sitive graph admits a perfect matching. The two results together imply that every Cayley graph admits an invariant random perfect matching. A key step in the proof is a result on graphings that also applies to finite graphs. The finite version says that for any partial matching of a finite regular graph that is a good expander, one can always find an augmenting path whose length is poly-logarithmic in one over the ratio of unmatched vertices.

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