{"title":"General sharp upper bounds on the total coalition number","authors":"J'anos Bar'at, Zolt'an L. Bl'azsik","doi":"10.7151/dmgt.2511","DOIUrl":null,"url":null,"abstract":"Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\\cup B$ is a total dominating set. A vertex partition $\\Psi=\\{C_1,C_2,\\dots,C_k\\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\\in\\{1,2,\\dots,k\\}$ there exists a distinct $j\\in\\{1,2,\\dots,k\\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2511","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition $\Psi=\{C_1,C_2,\dots,C_k\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\in\{1,2,\dots,k\}$ there exists a distinct $j\in\{1,2,\dots,k\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.