{"title":"论图积的k独立数","authors":"A. Abiad, Hidde Koerts","doi":"10.7151/dmgt.2480","DOIUrl":null,"url":null,"abstract":"Abstract The k-independence number of a graph, αk(G), is the maximum size of a set of vertices at pairwise distance greater than k, or alternatively, the independence number of the k-th power graph Gk. Although it is known that αk(G) = α(Gk), this, in general, does not hold for most graph products, and thus the existing bounds for α of graph products cannot be used. In this paper we present sharp upper bounds for the k-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for k = 1 follow as corollaries of our main results.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On the k-Independence Number of Graph Products\",\"authors\":\"A. Abiad, Hidde Koerts\",\"doi\":\"10.7151/dmgt.2480\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract The k-independence number of a graph, αk(G), is the maximum size of a set of vertices at pairwise distance greater than k, or alternatively, the independence number of the k-th power graph Gk. Although it is known that αk(G) = α(Gk), this, in general, does not hold for most graph products, and thus the existing bounds for α of graph products cannot be used. In this paper we present sharp upper bounds for the k-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for k = 1 follow as corollaries of our main results.\",\"PeriodicalId\":48875,\"journal\":{\"name\":\"Discussiones Mathematicae Graph Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-05-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discussiones Mathematicae Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2480\",\"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.2480","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract The k-independence number of a graph, αk(G), is the maximum size of a set of vertices at pairwise distance greater than k, or alternatively, the independence number of the k-th power graph Gk. Although it is known that αk(G) = α(Gk), this, in general, does not hold for most graph products, and thus the existing bounds for α of graph products cannot be used. In this paper we present sharp upper bounds for the k-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for k = 1 follow as corollaries of our main results.
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.