{"title":"1-平面图的线性拟合性","authors":"Weifan Wang, Juan Liu, Yiqiao Wang","doi":"10.7151/dmgt.2453","DOIUrl":null,"url":null,"abstract":"Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈ Δ(G)2 ⌉≤la(G)≤⌈ Δ(G)+12 ⌉ \\left\\lceil {{{\\Delta \\left( G \\right)} \\over 2}} \\right\\rceil \\le la\\left( G \\right) \\le \\left\\lceil {{{\\Delta \\left( G \\right) + 1} \\over 2}} \\right\\rceil for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Linear Arboricity of 1-Planar Graphs\",\"authors\":\"Weifan Wang, Juan Liu, Yiqiao Wang\",\"doi\":\"10.7151/dmgt.2453\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈ Δ(G)2 ⌉≤la(G)≤⌈ Δ(G)+12 ⌉ \\\\left\\\\lceil {{{\\\\Delta \\\\left( G \\\\right)} \\\\over 2}} \\\\right\\\\rceil \\\\le la\\\\left( G \\\\right) \\\\le \\\\left\\\\lceil {{{\\\\Delta \\\\left( G \\\\right) + 1} \\\\over 2}} \\\\right\\\\rceil for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13.\",\"PeriodicalId\":48875,\"journal\":{\"name\":\"Discussiones Mathematicae Graph Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discussiones Mathematicae Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2453\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discussiones Mathematicae Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2453","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈ Δ(G)2 ⌉≤la(G)≤⌈ Δ(G)+12 ⌉ \left\lceil {{{\Delta \left( G \right)} \over 2}} \right\rceil \le la\left( G \right) \le \left\lceil {{{\Delta \left( G \right) + 1} \over 2}} \right\rceil for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13.
期刊介绍:
The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.