{"title":"图的跨偶数树","authors":"Bill Jackson, Kiyoshi Yoshimoto","doi":"10.1002/jgt.23115","DOIUrl":null,"url":null,"abstract":"<p>A tree <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> is said to be <i>even</i> if all pairs of vertices of degree one in <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> are joined by a path of even length. We conjecture that every <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-regular nonbipartite connected graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has a spanning even tree and verify this conjecture when <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> is odd and <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has at least <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> bridges. We investigate this case further and propose some related conjectures.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Spanning even trees of graphs\",\"authors\":\"Bill Jackson, Kiyoshi Yoshimoto\",\"doi\":\"10.1002/jgt.23115\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A tree <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> is said to be <i>even</i> if all pairs of vertices of degree one in <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> are joined by a path of even length. We conjecture that every <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-regular nonbipartite connected graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> has a spanning even tree and verify this conjecture when <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math> is odd and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> has at least <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math> bridges. We investigate this case further and propose some related conjectures.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23115\",\"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://onlinelibrary.wiley.com/doi/10.1002/jgt.23115","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A tree is said to be even if all pairs of vertices of degree one in are joined by a path of even length. We conjecture that every -regular nonbipartite connected graph has a spanning even tree and verify this conjecture when has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when is odd and has at least bridges. We investigate this case further and propose some related conjectures.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
