{"title":"有符号图形笛卡尔积的边着色","authors":"","doi":"10.1016/j.disc.2024.114276","DOIUrl":null,"url":null,"abstract":"<div><div>According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1 if <em>G</em> is Class 1 or both <em>G</em> and <em>H</em> have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1 if one of the factors, say <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>, is Class 1 and there exists an edge coloring of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> have a Δ-matching and at least one of <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>Δ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is even, then <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1. This implies that if both of <em>G</em> and <em>H</em> have a Δ-matching, then <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1, thereby slightly improving upon Mohar's results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The edge coloring of the Cartesian product of signed graphs\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114276\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1 if <em>G</em> is Class 1 or both <em>G</em> and <em>H</em> have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1 if one of the factors, say <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>, is Class 1 and there exists an edge coloring of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> have a Δ-matching and at least one of <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>Δ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is even, then <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1. This implies that if both of <em>G</em> and <em>H</em> have a Δ-matching, then <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1, thereby slightly improving upon Mohar's results.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-10-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24004072\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004072","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
根据维京定理,边着色领域的一个主要问题是确定一个图是第 1 类还是第 2 类。1984 年,莫哈尔证明,如果 G 是第 1 类图,或者 G 和 H 都有完美匹配,则笛卡尔积 G□H 是第 1 类图。最近,Behr 证明了 Vizing 定理的有符号图版本:有符号图 (G,σ) 要么是第 1 类,要么是第 2 类。因此,我们希望将莫哈尔的结果推广到有符号图。在本文中,我们将证明,如果其中一个因子(比如说 (G,σ))是第 1 类,并且 (G,σ) 的边着色满足某个属性,则 (G,σ)□(H,π) 是第 1 类。假设 Δ-matching 是一个覆盖了最大度顶点的匹配。我们还证明,如果(G,σ)和(H,π)都有Δ匹配,并且Δ(G),Δ(H)中至少有一个是偶数,那么(G,σ)□(H,π)就是第 1 类。这意味着如果 G 和 H 都有Δ匹配,那么 G□H 是第 1 类,从而稍微改进了莫哈尔的结果。
The edge coloring of the Cartesian product of signed graphs
According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product is Class 1 if G is Class 1 or both G and H have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that is Class 1 if one of the factors, say , is Class 1 and there exists an edge coloring of that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of and have a Δ-matching and at least one of is even, then is Class 1. This implies that if both of G and H have a Δ-matching, then is Class 1, thereby slightly improving upon Mohar's results.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.