{"title":"关于完全多部排列图","authors":"A. S. Razafimahatratra","doi":"10.26493/1855-3974.2554.856","DOIUrl":null,"url":null,"abstract":"Given a finite transitive permutation group $G\\leq \\operatorname{Sym}(\\Omega)$, with $|\\Omega|\\geq 2$, the derangement graph $\\Gamma_G$ of $G$ is the Cayley graph $\\operatorname{Cay}(G,\\operatorname{Der}(G))$, where $\\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $\\operatorname{Sym}(2)$ acting on $\\{1,2\\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. \nThis paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"On complete multipartite derangement graphs\",\"authors\":\"A. S. Razafimahatratra\",\"doi\":\"10.26493/1855-3974.2554.856\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a finite transitive permutation group $G\\\\leq \\\\operatorname{Sym}(\\\\Omega)$, with $|\\\\Omega|\\\\geq 2$, the derangement graph $\\\\Gamma_G$ of $G$ is the Cayley graph $\\\\operatorname{Cay}(G,\\\\operatorname{Der}(G))$, where $\\\\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $\\\\operatorname{Sym}(2)$ acting on $\\\\{1,2\\\\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. \\nThis paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.\",\"PeriodicalId\":8402,\"journal\":{\"name\":\"Ars Math. Contemp.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-02-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Math. Contemp.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.2554.856\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2554.856","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $\operatorname{Sym}(2)$ acting on $\{1,2\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite.
This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.