Homomorphism Thresholds for Odd Cycles

Abstract

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph F we define the homomorphism threshold as the infimum over all α ∈ [0,1] such that every n-vertex F-free graph G with minimum degree at least αn has a homomorphic image H of bounded order (i.e. independent of n), which is F-free as well. Without the restriction of H being F-free we recover the definition of the chromatic threshold, which was determined for every graph F by Allen et al. [1]. The homomorphism threshold is less understood and we address the problem for odd cycles.

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Acknowledgments

We thank both referees for their detailed and helpful remarks.

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Correspondence to Oliver Ebsen or Mathias Schacht.

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The second author is supported by ERC Consolidator Grant 724903.

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Ebsen, O., Schacht, M. Homomorphism Thresholds for Odd Cycles. Combinatorica 40, 39–62 (2020). https://doi.org/10.1007/s00493-019-3920-8

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Mathematics Subject Classification (2010)

  • 05C35
  • 05C07
  • 05C15
  • 05D40