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The multiplication table problem for bipartite graphs

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Abstract

We investigate the following generalisation of the ‘multiplication table problem’ of Erdős: given a bipartite graph with m edges, how large is the set of sizes of its induced subgraphs? Erdős’s problem of estimating the number of distinct products ab with a,b ≤ n is precisely the problem under consideration when the graph in question is the complete bipartite graph K n,n . In this note, we prove that the set of sizes of the induced subgraphs of any bipartite graph with m edges contains Ω(m/(logm)12) distinct elements.

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB30WB, UK

    Bhargav P. Narayanan & István Tomon

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Julian Sahasrabudhe

Authors
  1. Bhargav P. Narayanan
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  2. Julian Sahasrabudhe
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  3. István Tomon
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Correspondence to Bhargav P. Narayanan.

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Narayanan, B.P., Sahasrabudhe, J. & Tomon, I. The multiplication table problem for bipartite graphs. Combinatorica 37, 991–1010 (2017). https://doi.org/10.1007/s00493-016-3322-0

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  • DOI: https://doi.org/10.1007/s00493-016-3322-0

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