Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\overrightarrow{C_{k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <msub> <mi>C</mi> <mi>k</mi> </msub> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> be the transitive cycle on k vertices. Motivated by work of Balogh, Lo, and Molla [J. Combin. Theory Ser.B 124:64–87, 2017] showed that every n-vertex oriented graph D with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\in 3\mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mn>3</mn> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta ^{0}(D)\ge 7n/18$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>δ</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>7</mn> <mi>n</mi> <mo stretchy="false">/</mo> <mn>18</mn> </mrow> </math></EquationSource> </InlineEquation> contains a <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\overrightarrow{C_{3}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>-factor. We consider the next open subcase, and show that every n-vertex oriented graph D with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\in 4\mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mn>4</mn> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\delta (D)\ge 3n/4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>3</mn> <mi>n</mi> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> (or <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta ^{0}(D)\ge 3n/8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>δ</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>3</mn> <mi>n</mi> <mo stretchy="false">/</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>) contains a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\overrightarrow{C_4}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <msub> <mi>C</mi> <mn>4</mn> </msub> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>-factor. These two bounds are tight respectively.

A note between transitive $C_4$-factor and oriented Ramsey number

期刊论文
发表日期：

2026年2月2日

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Ming Chen,
Zhengke Miao,
Shan Zhou

摘要

Let $\overrightarrow{C_{k}}$ $\vec{C_{k}}$ be the transitive cycle on k vertices. Motivated by work of Balogh, Lo, and Molla [J. Combin. Theory Ser.B 124:64–87, 2017] showed that every n-vertex oriented graph D with $n\in 3\mathbb {N}$ $n \in 3 N$ and $\delta ^{0}(D)\ge 7n/18$ $δ^{0} (D) \geq 7 n / 18$ contains a $\overrightarrow{C_{3}}$ $\vec{C_{3}}$ -factor. We consider the next open subcase, and show that every n-vertex oriented graph D with $n\in 4\mathbb {N}$ $n \in 4 N$ and $\delta (D)\ge 3n/4$ $δ (D) \geq 3 n / 4$ (or $\delta ^{0}(D)\ge 3n/8$ $δ^{0} (D) \geq 3 n / 8$ ) contains a $\overrightarrow{C_4}$ $\vec{C_{4}}$ -factor. These two bounds are tight respectively.

A note between transitive \(C_4\)-factor and oriented Ramsey number

摘要