![PDF Download](/rc-pub/front/front-article/download/176040563/lowqualitypdf/A Note on Rainbow <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mi mathvariant="bold-italic">k</mi></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/CBC851C6-5ED6-4fc3-800B-4F3656A3A27A-M001.png"><?fx-imagestate width="4.37444448" height="6.06777763"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/CBC851C6-5ED6-4fc3-800B-4F3656A3A27A-M001c.png"><?fx-imagestate width="4.37444448" height="6.06777763"?></graphic></alternatives></inline-formula>-Factor with Large Edge Numbers.pdf){kind=link}
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Mathematics | 更新时间：2026-10-29
|
A Note on Rainbow
- Wuhan University Journal of Natural Sciences Vol. 31, Issue 5, Pages: 443-447(2026)
- 作者机构：
西北师范大学 数学与统计学院， 甘肃 兰州 730070
- 作者简介：
WEN Ningning, female, Master candidate, research direction: graph theory. E-mail: xianing0110@163.com
E-mail: liuxia_90@163.com
- 基金信息：
DOI：10.1051/wujns/2026315443
CLC： O157.5Received：23 August 2025，
Published：24 October 2026
- 稿件说明：
移动端阅览
文宁宁,刘霞.具有较大边数的图集的彩虹k-因子[J].武汉大学学报自然科学英文版,2026,31(05):443-447.
WEN Ningning, LIU Xia. A Note on Rainbow k-Factor with Large Edge Numbers[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 443-447.
文宁宁,刘霞.具有较大边数的图集的彩虹k-因子[J].武汉大学学报自然科学英文版,2026,31(05):443-447. DOI： 10.1051/wujns/2026315443.
WEN Ningning, LIU Xia. A Note on Rainbow k-Factor with Large Edge Numbers[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 443-447. DOI： 10.1051/wujns/2026315443.
A Note on Rainbow ![]()
![]()
摘要
对整数
<math id="M18"><mi>k</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275334&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275319&type=
6.91444445
2.68111110
与
<math id="M19"><mi>n</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275363&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275362&type=
11.99444389
2.68111110
（
<math id="M20"><mi>k</mi><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275323&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275338&type=
3.52777815
2.68111110
为偶数），设
<math id="M21"><mi mathvariant="italic">𝒢</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>
</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>
</mo><mo>⋯</mo><mo>
</mo><mtext> </mtext><msub><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msub><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275353&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275340&type=
30.05666542
6.91444445
为具有相同顶点集合
<math id="M22"><mi>V</mi><mo>=</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275368&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275326&type=
9.03111172
3.38666677
的图集。如果
<math id="M23"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275404&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275374&type=
2.39888883
3.10444450
有一个
<math id="M24"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275371&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275370&type=
1.83444452
2.68111110
-正则图，其每条边来自
<math id="M25"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275418&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275380&type=
2.39888883
3.10444450
中不同的图，则称该子图为
<math id="M26"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275418&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275380&type=
2.39888883
3.10444450
的一个彩虹
<math id="M27"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275410&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275419&type=
1.83444452
2.68111110
-因子。在本文中，我们证明了如果对每个
<math id="M28"><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">1</mn><mo>
</mo><mtext> </mtext><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">]</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275423&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275385&type=
15.09888935
7.19666624
，都有图
<math id="M29"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275426&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275400&type=
3.52777815
3.80999994
的边数
<math id="M30"><mi>ε</mi><mo stretchy="false">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275434&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275433&type=
7.61999989
4.51555586
满足
<math id="M31"><mi>ε</mi><mo stretchy="false">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275451&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275450&type=
31.89110947
7.19666624
，那么
<math id="M32"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275418&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275380&type=
2.39888883
3.10444450
有一个彩虹
<math id="M33"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275410&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275419&type=
1.83444452
2.68111110
-因子。特别地，我们还刻画了最小度至少为2的图集具有彩虹2-因子的边数条件。
Abstract
For integers
<math id="M3"><mi>k</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275246&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275207&type=
5.92666674
2.25777793
<math id="M4"><mi>n</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275220&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275247&type=
10.30111122
2.25777793
with
<math id="M5"><mi>k</mi><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275222&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275221&type=
3.10444450
2.25777793
even
let
<math id="M6"><mi mathvariant="italic">𝒢</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>
</mo><mtext> </mtext><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>
</mo><mo>⋯</mo><mo>
</mo><mtext> </mtext><msub><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msub><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275283&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275269&type=
26.52888680
6.06777763
be a family of graphs on the common vertex set
<math id="M7"><mi>V</mi><mo>=</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275273&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275285&type=
7.76111078
2.82222223
. We say
<math id="M8"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275276&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275262&type=
2.11666679
2.68111110
admits a rainbow
<math id="M9"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275265&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275290&type=
1.55222225
2.25777793
-factor if there exists a
<math id="M10"><mtext> </mtext><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275313&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275280&type=
2.25777793
2.39888883
-regular subgraph of the union of graphs in
<math id="M11"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275276&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275262&type=
2.11666679
2.68111110
such that all edges of the subgraph come from different members of
<math id="M12"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275276&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275262&type=
2.11666679
2.68111110
. In this paper
we prove that if the edge number
<math id="M13"><mi>ε</mi><mo stretchy="false">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275310&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275295&type=
27.37555313
6.20888901
for each
<math id="M14"><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">1</mn><mo>
</mo><mtext> </mtext><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">]</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275332&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275311&type=
12.98222256
6.20888901
then
<math id="M15"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275276&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275262&type=
2.11666679
2.68111110
admits a rainbow
<math id="M16"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275265&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275290&type=
1.55222225
2.25777793
-factor. In particular
we also characterize the edge number condition for a rainbow 2-factor when each graph in
<math id="M17"><mi mathvariant="italic">𝒢</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275276&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275262&type=
2.11666679
2.68111110
has minimum degree at least 2.
关键词
Keywords
references
Bondy J A , Murty U S R . Graph Theory ( Graduate Texts in Mathematics ) [M ] . New York : Springer-Verlag , 2008 .
Brouwer A E , Haemers W H . Eigenvalues and perfect matchings [J ] . Linear Algebra and Its Applications , 2005 , 395 ( 1 ): 155 - 162 .
Cioabă S M . Perfect matchings, eigenvalues and expansion [J ] . Mathematical Reports of the Academy of Science of the Royal Society of Canada , 2005 , 27 ( 4 ): 101 - 104 .
Cioabă S M , Gregory D A . Large matchings from eigenvalues [J ] . Linear Algebra and Its Applications , 2007 , 422 ( 1 ): 308 - 317 .
Cioabă S M , Gregory D A , Haemers W H . Matchings in regular graphs from eigenvalues [J ] . Journal of Combinatorial Theory, Series B , 2009 , 99 ( 2 ): 287 - 297 .
Lu H . Regular factors of regular graphs from eigenvalues [J ] . The Electronic Journal of Combinatorics , 2010 , 17 ( 1 ): 159 .
Lu H . Regular graphs, eigenvalues and regular factors [J ] . Journal of Graph Theory , 2012 , 69 ( 4 ): 349 - 355 .
Fan D , Lin H , Lu H . Spectral radius and <math id="M307"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121277204&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121277168&type= 5.78555536 2.82222223 -factors in graphs [J ] . Discrete Mathematics , 2022 , 345 ( 7 ): 112892 .
Cho E K , Hyun J Y , Suil O , Park J R . Sharp conditions for the existence of an even <math id="M308"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121277204&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121277168&type= 5.78555536 2.82222223 -factor in a graph [J ] . Bulletin of the Korean Mathematical Society , 2021 , 58 ( 1 ): 31 - 46 .
Zhang L , Zhang Z . Spectral radius and rainbow <math id="M309"><mi>k</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275440&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121275439&type= 1.55222225 2.25777793 -factor of graphs [J ] . Linear Algebra and Its Applications , 2026 , 735 : 307 - 318 .
Kelmans A K . On graphs with randomly deleted edges [J ] . Acta Mathematica Academiae Scientiarum Hungarica , 1981 , 37 ( 1-4 ): 77 - 88 .
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