Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {gr}_k(G:H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>gr</mo> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is defined as the minimum positive integer N such that for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, every k-edge-coloring of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> contains either a rainbow subgraph G or a monochromatic subgraph H. In this paper, we get some exact values or bounds of Gallai–Ramsey numbers for rainbow <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$K_{1,3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$P_5$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_4^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>P</mi> <mn>4</mn> <mo>+</mo> </msubsup> </math></EquationSource> </InlineEquation> and monochromatic double stars or books.

Gallai–Ramsey numbers for rainbow trees and monochromatic double stars or books

期刊论文
发表日期：

2025年1月17日

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Yuan Si

摘要

Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number $\operatorname {gr}_k(G:H)$ ${gr}_{k} (G : H)$ is defined as the minimum positive integer N such that for all $n\ge N$ $n \geq N$ , every k-edge-coloring of $K_n$ $K_{n}$ contains either a rainbow subgraph G or a monochromatic subgraph H. In this paper, we get some exact values or bounds of Gallai–Ramsey numbers for rainbow $K_{1,3}$ $K_{1, 3}$ , $P_5$ $P_{5}$ or $P_4^{+}$ $P_{4}^{+}$ and monochromatic double stars or books.