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The Chromatic Number of the Product of 5-Chromatic Graphs can be 4

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Abstract

We show that for any integers \(m_1, m_2 \ge 5\), there exist graphs G, H such that \(\chi (G) = m_1\), \(\chi (H) = m_2\) and \(\chi (G \times H) < \min \{m_1, m_2\}\). In particular for \(m_1 = m_2 = 5\), this settles the last remaining open case of Hedetniemi’s conjecture.

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Notes

  1. I am indebted to Gábor Tardos for a suggestion that led to a simplification of this Lemma and its proof.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada

    Claude Tardif

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  1. Claude Tardif
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Correspondence to Claude Tardif.

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Tardif, C. The Chromatic Number of the Product of 5-Chromatic Graphs can be 4. Combinatorica 43, 1067–1073 (2023). https://doi.org/10.1007/s00493-023-00047-2

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