4-coverable snarks, perfect matching cover, and Isaacs product

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2025-02-07 DOI:10.1016/j.dam.2025.02.001

Wenjuan Zhou , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang

{"title":"4-coverable snarks, perfect matching cover, and Isaacs product","authors":"Wenjuan Zhou , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang","doi":"10.1016/j.dam.2025.02.001","DOIUrl":null,"url":null,"abstract":"<div><div>The perfect matching cover index <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a bridgeless cubic graph <span><math><mi>G</mi></math></span> is the minimum number of perfect matchings in <span><math><mi>G</mi></math></span> that cover all edges of <span><math><mi>G</mi></math></span>. It is conjectured by Berge that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>5</mn></mrow></math></span> for every bridgeless cubic graph. Esperet and Mazzuoccolo (2013) proved that deciding whether a cubic bridgeless graph <span><math><mi>G</mi></math></span> satisfies <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>4</mn></mrow></math></span> is NP-complete. Some major conjectures hold for bridgeless cubic graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>, such as the cycle double cover conjecture and the Alon–Tarsi shortest cycle cover conjecture. In this paper, we study the perfect matching cover index of snarks obtained by the Isaacs products of two cubic graphs. Our results indicate that the Isaacs product of certain snarks does not increase the perfect matching cover index. As corollaries, we show that the perfect matching cover indices of some families of snarks with arbitrarily large oddness equal to 4.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 68-79"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25000599","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

The perfect matching cover index

τ (G)

of a bridgeless cubic graph

G

is the minimum number of perfect matchings in

G

that cover all edges of

G

. It is conjectured by Berge that

τ (G) \leq 5

for every bridgeless cubic graph. Esperet and Mazzuoccolo (2013) proved that deciding whether a cubic bridgeless graph

G

satisfies

τ (G) \leq 4

is NP-complete. Some major conjectures hold for bridgeless cubic graphs

G

with

τ (G) = 4

, such as the cycle double cover conjecture and the Alon–Tarsi shortest cycle cover conjecture. In this paper, we study the perfect matching cover index of snarks obtained by the Isaacs products of two cubic graphs. Our results indicate that the Isaacs product of certain snarks does not increase the perfect matching cover index. As corollaries, we show that the perfect matching cover indices of some families of snarks with arbitrarily large oddness equal to 4.

查看原文本刊更多论文

求助全文

约1分钟内获得全文求助全文

来源期刊

Discrete Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.