关于Alon-Tarsi最短周期覆盖猜想的注记

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2026-04-13 Epub Date: 2026-03-20 DOI:10.1002/jgt.70026

Yilun Luo, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang

{"title":"关于Alon-Tarsi最短周期覆盖猜想的注记","authors":"Yilun Luo, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang","doi":"10.1002/jgt.70026","DOIUrl":null,"url":null,"abstract":"<div>\n \n The shortest cycle cover conjecture (SCC conjecture), proposed by Alon and Tarsi, asserts that every bridgeless cubic graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> has a cycle cover with a total length at most <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mn>21</mn>\n \n <mn>15</mn>\n </mfrac>\n \n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n </semantics></math>. Tarsi further proposed a related conjecture, the <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>SCC</mi>\n \n <mn>3</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math> conjecture, which states that every bridgeless cubic graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> has a 3-cycle cover with a total length at most <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mn>22</mn>\n \n <mn>15</mn>\n </mfrac>\n \n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n </semantics></math>. In this paper, we prove that every cyclically odd-<math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mn>4</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math>-edge-connected cubic graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> has a 3-cycle cover with a total length at most <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mn>20</mn>\n \n <mn>15</mn>\n </mfrac>\n \n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>+</mo>\n \n <mfrac>\n <mn>1</mn>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mfrac>\n \n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n </semantics></math>. Consequently, the SCC conjecture and the <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>SCC</mi>\n \n <mn>3</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math> conjecture are verified in this paper for cubic graphs with cyclic odd-edge-connectivity at least 29 and 17, respectively. Additionally, for cubic graphs that satisfies Kaiser–Raspaud conjecture (i.e., every bridgeless cubic graph has two perfect matchings <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math> and a parity subgraph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>J</mi>\n </mrow>\n </mrow>\n </semantics></math>, such that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>∩</mo>\n \n <msub>\n <mi>M</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>∩</mo>\n \n <mi>J</mi>\n \n <mo>=</mo>\n \n <mo>∅</mo>\n </mrow>\n </mrow>\n </semantics></math>), the SCC conjecture and the <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>SCC</mi>\n \n <mn>3</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math> conjecture are verified for graphs with girth at least 20 and 10, respectively.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"172-177"},"PeriodicalIF":1.0000,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Note on Alon–Tarsi Shortest Cycle Cover Conjecture\",\"authors\":\"Yilun Luo, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang\",\"doi\":\"10.1002/jgt.70026\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n The shortest cycle cover conjecture (SCC conjecture), proposed by Alon and Tarsi, asserts that every bridgeless cubic graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> has a cycle cover with a total length at most <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mfrac>\\n <mn>21</mn>\\n \\n <mn>15</mn>\\n </mfrac>\\n \\n <mo>∣</mo>\\n \\n <mi>E</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n </mrow>\\n </mrow>\\n </semantics></math>. Tarsi further proposed a related conjecture, the <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>SCC</mi>\\n \\n <mn>3</mn>\\n </msub>\\n </mrow>\\n </mrow>\\n </semantics></math> conjecture, which states that every bridgeless cubic graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> has a 3-cycle cover with a total length at most <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mfrac>\\n <mn>22</mn>\\n \\n <mn>15</mn>\\n </mfrac>\\n \\n <mo>∣</mo>\\n \\n <mi>E</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n </mrow>\\n </mrow>\\n </semantics></math>. In this paper, we prove that every cyclically odd-<math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mn>4</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math>-edge-connected cubic graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> has a 3-cycle cover with a total length at most <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mfrac>\\n <mn>20</mn>\\n \\n <mn>15</mn>\\n </mfrac>\\n \\n <mo>∣</mo>\\n \\n <mi>E</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n \\n <mo>+</mo>\\n \\n <mfrac>\\n <mn>1</mn>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mfrac>\\n \\n <mo>∣</mo>\\n \\n <mi>E</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n </mrow>\\n </mrow>\\n </semantics></math>. Consequently, the SCC conjecture and the <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>SCC</mi>\\n \\n <mn>3</mn>\\n </msub>\\n </mrow>\\n </mrow>\\n </semantics></math> conjecture are verified in this paper for cubic graphs with cyclic odd-edge-connectivity at least 29 and 17, respectively. Additionally, for cubic graphs that satisfies Kaiser–Raspaud conjecture (i.e., every bridgeless cubic graph has two perfect matchings <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>M</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n </mrow>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>M</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n </mrow>\\n </semantics></math> and a parity subgraph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>J</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>, such that <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>M</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>∩</mo>\\n \\n <msub>\\n <mi>M</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mo>∩</mo>\\n \\n <mi>J</mi>\\n \\n <mo>=</mo>\\n \\n <mo>∅</mo>\\n </mrow>\\n </mrow>\\n </semantics></math>), the SCC conjecture and the <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>SCC</mi>\\n \\n <mn>3</mn>\\n </msub>\\n </mrow>\\n </mrow>\\n </semantics></math> conjecture are verified for graphs with girth at least 20 and 10, respectively.\\n </div>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"112 2\",\"pages\":\"172-177\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2026-04-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.70026\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2026/3/20 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.70026","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/3/20 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

由Alon和Tarsi提出的最短周期覆盖猜想（SCC猜想）断言每一个无桥三次图G都有一个总长度不超过2115∣E的环盖(g)∣。Tarsi进一步提出了一个相关的猜想，scc3猜想，它说明每个无桥三次图G都有一个总长度不超过22 15∣E(g)∣。在本文中，我们证明了每一个循环奇- （4k + 1）-边连通三次图G有一个总长度不超过2015∣E的3环盖(G)∣+ 12k+ 1∣e (g)∣。因此，本文分别对循环奇边连通数至少为29和17的三次图验证了SCC猜想和SCC 3猜想。此外，对于满足Kaiser-Raspaud猜想的三次图(即 ,每个无桥三次图都有两个完美匹配m1和m2和奇偶子图J，使得m1∩m2∩J =∅)，分别对周长至少为20和10的图验证SCC猜想和SCC 3猜想。

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A Note on Alon–Tarsi Shortest Cycle Cover Conjecture

The shortest cycle cover conjecture (SCC conjecture), proposed by Alon and Tarsi, asserts that every bridgeless cubic graph $G$ has a cycle cover with a total length at most $\frac{21}{15} ∣ E (G) ∣$ . Tarsi further proposed a related conjecture, the ${SCC}_{3}$ conjecture, which states that every bridgeless cubic graph $G$ has a 3-cycle cover with a total length at most $\frac{22}{15} ∣ E (G) ∣$ . In this paper, we prove that every cyclically odd- $(4 k + 1)$ -edge-connected cubic graph $G$ has a 3-cycle cover with a total length at most $\frac{20}{15} ∣ E (G) ∣ + \frac{1}{2 k + 1} ∣ E (G) ∣$ . Consequently, the SCC conjecture and the ${SCC}_{3}$ conjecture are verified in this paper for cubic graphs with cyclic odd-edge-connectivity at least 29 and 17, respectively. Additionally, for cubic graphs that satisfies Kaiser–Raspaud conjecture (i.e., every bridgeless cubic graph has two perfect matchings $M_{1}$ and $M_{2}$ and a parity subgraph $J$ , such that $M_{1} \cap M_{2} \cap J = \emptyset$ ), the SCC conjecture and the ${SCC}_{3}$ conjecture are verified for graphs with girth at least 20 and 10, respectively.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .