{"title":"将周长为 5 的平面图划分为两个最大度数为 4 的森林","authors":"Min Chen, André Raspaud, Weifan Wang, Weiqiang Yu","doi":"10.21136/cmj.2024.0394-21","DOIUrl":null,"url":null,"abstract":"<p>Given a graph <i>G</i> = (<i>V, E</i>), if we can partition the vertex set <i>V</i> into two nonempty subsets <i>V</i><sub>1</sub> and <i>V</i><sub>2</sub> which satisfy Δ(<i>G</i>[<i>V</i><sub>1</sub>]) ⩽ <i>d</i><sub>1</sub> and Δ(<i>G</i>[<i>V</i><sub>2</sub>]) ⩽ <i>d</i><sub>2</sub>, then we say <i>G</i> has a (<span>\\({{\\rm{\\Delta }}_{{d_1}}}\\,,{{\\rm{\\Delta }}_{{d_2}}}\\)</span>)-partition. And we say <i>G</i> admits an (<span>\\({F_{d_{1}}}, {F_{d_{2}}}\\)</span>)-partition if <i>G</i>[<i>V</i><sub>1</sub>] and <i>G</i>[<i>V</i><sub>2</sub>] are both forests whose maximum degree is at most <i>d</i><sub>1</sub> and <i>d</i><sub>2</sub>, respectively. We show that every planar graph with girth at least 5 has an (<i>F</i><sub>4</sub>, <i>F</i><sub>4</sub>)-partition.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Partitioning planar graph of girth 5 into two forests with maximum degree 4\",\"authors\":\"Min Chen, André Raspaud, Weifan Wang, Weiqiang Yu\",\"doi\":\"10.21136/cmj.2024.0394-21\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a graph <i>G</i> = (<i>V, E</i>), if we can partition the vertex set <i>V</i> into two nonempty subsets <i>V</i><sub>1</sub> and <i>V</i><sub>2</sub> which satisfy Δ(<i>G</i>[<i>V</i><sub>1</sub>]) ⩽ <i>d</i><sub>1</sub> and Δ(<i>G</i>[<i>V</i><sub>2</sub>]) ⩽ <i>d</i><sub>2</sub>, then we say <i>G</i> has a (<span>\\\\({{\\\\rm{\\\\Delta }}_{{d_1}}}\\\\,,{{\\\\rm{\\\\Delta }}_{{d_2}}}\\\\)</span>)-partition. And we say <i>G</i> admits an (<span>\\\\({F_{d_{1}}}, {F_{d_{2}}}\\\\)</span>)-partition if <i>G</i>[<i>V</i><sub>1</sub>] and <i>G</i>[<i>V</i><sub>2</sub>] are both forests whose maximum degree is at most <i>d</i><sub>1</sub> and <i>d</i><sub>2</sub>, respectively. We show that every planar graph with girth at least 5 has an (<i>F</i><sub>4</sub>, <i>F</i><sub>4</sub>)-partition.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.21136/cmj.2024.0394-21\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/cmj.2024.0394-21","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
给定一个图 G = (V，E)，如果我们可以将顶点集 V 分成两个非空子集 V1 和 V2，且这两个子集满足 Δ(G[V1]) ⩽ d1 和 Δ(G[V2]) ⩽ d2、那么我们说 G 有一个 (\({{\rm{\Delta }}_{d_1}}}\,,{{\rm{\Delta }}_{d_2}}\))-partition.如果 G[V1] 和 G[V2] 都是最大度分别最多为 d1 和 d2 的森林，我们就说 G 有一个 (\({F_{d_{1}}}, {F_{d_{2}}}))-分区。我们证明，每个周长至少为 5 的平面图都有一个 (F4, F4) 分离。
Partitioning planar graph of girth 5 into two forests with maximum degree 4
Given a graph G = (V, E), if we can partition the vertex set V into two nonempty subsets V1 and V2 which satisfy Δ(G[V1]) ⩽ d1 and Δ(G[V2]) ⩽ d2, then we say G has a (\({{\rm{\Delta }}_{{d_1}}}\,,{{\rm{\Delta }}_{{d_2}}}\))-partition. And we say G admits an (\({F_{d_{1}}}, {F_{d_{2}}}\))-partition if G[V1] and G[V2] are both forests whose maximum degree is at most d1 and d2, respectively. We show that every planar graph with girth at least 5 has an (F4, F4)-partition.
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