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arXiv · 1102.0712

Matchings on infinite graphs

Charles Bordenave·Marc Lelarge·Justin Salez
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Abstract

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdös-Rényi random graphs.

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Charles Bordenave, Marc Lelarge, Justin Salez. 2012-04-11. Matchings on infinite graphs. https://arxiv.org/abs/1102.0712

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