Marcel Celaya, Kelly Choo, G. MacGillivray, K. Seyffarth
{"title":"Reconfiguring k-colourings of Complete Bipartite Graphs","authors":"Marcel Celaya, Kelly Choo, G. MacGillivray, K. Seyffarth","doi":"10.5666/KMJ.2016.56.3.647","DOIUrl":null,"url":null,"abstract":"Let H be a graph, and k ≥ χ(H) an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer k0(H) such that H has a cyclic Gray code of its k-colourings for all k ≥ k0(H). For complete bipartite graphs, we prove that k0(K`,r) = 3 when both ` and r are odd, and k0(K`,r) = 4 otherwise.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"647-655"},"PeriodicalIF":0.6000,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kyungpook Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5666/KMJ.2016.56.3.647","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 10
Abstract
Let H be a graph, and k ≥ χ(H) an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer k0(H) such that H has a cyclic Gray code of its k-colourings for all k ≥ k0(H). For complete bipartite graphs, we prove that k0(K`,r) = 3 when both ` and r are odd, and k0(K`,r) = 4 otherwise.
期刊介绍:
Kyungpook Mathematical Journal is an international journal devoted to significant research concerning all aspects of mathematics. The journal has a preference for papers having a broad interest. One volume of the journal is published every year. Each volume until volume 42 consisted of two issues; however, starting from volume 43(2003), each volume consists of four issues. Authors should strive for expository clarity and good literary style. Manuscripts should be prepared as follows. The first page must consist of a short descriptive title, followed by the name(s) and address(es) of the author(s) along with an electronic address if available.