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Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant

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Abstract.

It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant \( h(G) \geq k \) has a spanning forest in which each component is isomorphic to T(k).

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Authors and Affiliations

  1. Itai Benjamini and Oded Schramm, Department of Mathematics, The Weizmann Institute, Rehovot 76100, Israel, e-mails: itai@wisdom.weizmann.ac.il, schramm@wisdom.weizmann.ac.il, , , , , , IL

    I. Benjamini & O. Schramm

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  1. I. Benjamini
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  2. O. Schramm
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Submitted: September 1996, final version: October 1996

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Benjamini, I., Schramm, O. Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant. GAFA, Geom. funct. anal. 7, 403–419 (1997). https://doi.org/10.1007/PL00001625

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  • DOI: https://doi.org/10.1007/PL00001625

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