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<p>An edge colored graph is a rainbow if all colors on its edges are distinct. For two graphs <i>G</i> and <i>H</i>, where <i>G</i> contains <i>H</i> as a subgraph, the anti-Ramsey number of <i>H</i> in <i>G</i>, denoted by <i>AR</i>(<i>G, H</i>), is the largest integer <i>k</i> such that there exists a <i>k</i>-edge-coloring of <i>G</i> containing no rainbow <i>H</i>. Let <i>kC</i><sub>3</sub> denote the union of <i>k</i> independent triangles. The anti-Ramsey problem for cycles (including independent cycles) in a complete graph <i>K</i><sub><i>n</i></sub> has been studied well. We consider the problem for independent cycles in a tripartite graph and obtain the value of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(AR(K_{q_1,q_2,q_3},2C_3)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>A</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> for <i>q</i><sub>1</sub> ⩾ <i>q</i><sub>2</sub> ⩾ <i>q</i><sub>3</sub> ⩾ 2.</p>

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The anti-Ramsey problem for independent triangles in tripartite graphs

  • 期刊论文

  • 发表日期：

    2025年11月7日

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  • Zemin Jin,
  • Huifang Liu,
  • Qian Wang,
  • Zhenxin Cao

摘要

An edge colored graph is a rainbow if all colors on its edges are distinct. For two graphs G and H, where G contains H as a subgraph, the anti-Ramsey number of H in G, denoted by AR(G, H), is the largest integer k such that there exists a k-edge-coloring of G containing no rainbow H. Let kC3 denote the union of k independent triangles. The anti-Ramsey problem for cycles (including independent cycles) in a complete graph Kn has been studied well. We consider the problem for independent cycles in a tripartite graph and obtain the value of \(AR(K_{q_1,q_2,q_3},2C_3)\) A R ( K q 1 , q 2 , q 3 , 2 C 3 ) for q1 ⩾ q2 ⩾ q3 ⩾ 2.