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<p>Given two non-empty graphs <i>G, H</i> and a positive integer <i>k</i>, the Gallai-Ramsey number gr<sub><i>k</i></sub>(<i>G</i>: <i>H</i>) is defined as the minimum integer <i>N</i> such that for all <i>n</i> ≥ <i>N</i>, every exact <i>k</i>-edge-coloring of <i>K</i><sub><i>n</i></sub> contains either a rainbow copy of <i>G</i> or a monochromatic copy of <i>H</i>. Denote gr<sub><i>k</i></sub>′(<i>G</i>: <i>H</i>) as the minimum integer <i>N</i> such that for all <i>n</i> ≥ <i>N</i>, every edge-coloring of <i>K</i><sub><i>n</i></sub> using at most <i>k</i> colors contains either a rainbow copy of <i>G</i> or a monochromatic copy of <i>H</i>. In this paper, we get some exact values or bounds for gr<sub><i>k</i></sub>(<i>P</i><sub>5</sub>: <i>H</i>) and gr<sub><i>k</i></sub>′(<i>P</i><sub>5</sub>: <i>H</i>), where <i>H</i> is a cycle or a book graph. In addition, our results support a conjecture of Li, Besse, Magnant, Wang and Watts in 2020.</p>

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Ramsey and Gallai-Ramsey Numbers of Cycles and Books

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • Mei-qin Wei,
  • Ya-ping Mao,
  • Ingo Schiermeyer,
  • Zhao Wang

摘要

Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number grk(G: H) is defined as the minimum integer N such that for all n ≥ N, every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. Denote grk′(G: H) as the minimum integer N such that for all n ≥ N, every edge-coloring of Kn using at most k colors contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we get some exact values or bounds for grk(P5: H) and grk′(P5: H), where H is a cycle or a book graph. In addition, our results support a conjecture of Li, Besse, Magnant, Wang and Watts in 2020.