The multiplication table problem for bipartite graphs

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,bn 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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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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Mathematics Subject Classification (2000)

  • 05C35
  • 11B30