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<p>A Gallai <i>k</i>-coloring is a <i>k</i>-edge-coloring of a complete graph in which there are no rainbow triangles. For given graphs <i>G</i><sub>1</sub>, <i>G</i><sub>2</sub>, <i>G</i><sub>3</sub> and nonnegative integers <i>r, s, t</i> with <i>k</i> = <i>r</i> + <i>s</i> + <i>t</i>, the <i>k</i>-colored Gallai-Ramsey number <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>r</i>·<i>G</i><sub>1</sub>, <i>s</i>·<i>G</i><sub>2</sub>, <i>t</i>·<i>G</i><sub>3</sub>) is the minimum integer <i>n</i> such that every Gallai <i>k</i>-colored <i>K</i><sub><i>n</i></sub> contains a monochromatic copy of <i>G</i><sub>1</sub> colored by one of the first <i>r</i> colors or a monochromatic copy of <i>G</i><sub>2</sub> colored by one of the middle <i>s</i> colors or a monochromatic copy of <i>G</i><sub>3</sub> colored by one of the last <i>t</i> colors. In this paper, we determine the value of Gallai-Ramsey number in the case that <i>G</i><sub>1</sub> = <i>B</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>, <i>G</i><sub>2</sub> = <i>S</i><Stack> <sub>3</sub> <sup>+</sup> </Stack> and <i>G</i><sub>3</sub> = <i>K</i><sub>3</sub>. Then the Gallai-Ramsey numbers <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>B</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>), <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>S</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>) and <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>K</i><sub>3</sub>) are obtained, respectively. Furthermore, the Gallai-Ramsey numbers <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>r</i> · <i>B</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>, (<i>k</i> − <i>r</i>) · <i>S</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>), <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>r</i> · <i>B</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>, (<i>k</i> − <i>r</i>) · <i>K</i><sub>3</sub>) and <i>gr</i><sub><i>k</i></sub>(<i>K</i><sub>3</sub>: <i>s</i> · <i>S</i><Stack> <sub>3</sub> <sup>+</sup> </Stack>, (<i>k</i> − <i>s</i>) · <i>K</i><sub>3</sub>) are obtained, respectively.</p>

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Gallai-Ramsey numbers for three graphs on at most five vertices

  • 期刊论文

  • 发表日期：

    2025年3月25日

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  • Xue-li Su,
  • Yan Liu

摘要

A Gallai k-coloring is a k-edge-coloring of a complete graph in which there are no rainbow triangles. For given graphs G1, G2, G3 and nonnegative integers r, s, t with k = r + s + t, the k-colored Gallai-Ramsey number grk(K3: r·G1, s·G2, t·G3) is the minimum integer n such that every Gallai k-colored Kn contains a monochromatic copy of G1 colored by one of the first r colors or a monochromatic copy of G2 colored by one of the middle s colors or a monochromatic copy of G3 colored by one of the last t colors. In this paper, we determine the value of Gallai-Ramsey number in the case that G1 = B 3 + , G2 = S 3 + and G3 = K3. Then the Gallai-Ramsey numbers grk(K3: B 3 + ), grk(K3: S 3 + ) and grk(K3: K3) are obtained, respectively. Furthermore, the Gallai-Ramsey numbers grk(K3: r · B 3 + , (k − r) · S 3 + ), grk(K3: r · B 3 + , (k − r) · K3) and grk(K3: s · S 3 + , (k − s) · K3) are obtained, respectively.