{"title":"Shortening Array Codes and the Perfect 1-Factorization Conjecture","authors":"V. Bohossian, Jehoshua Bruck","doi":"10.1109/ISIT.2006.261572","DOIUrl":null,"url":null,"abstract":"The existence of a perfect 1-factorization of the complete graph K n, for arbitrary n, is a 40-year old open problem in graph theory. Two infinite families of perfect 1-factorizations are known for K2p and Kp+1, where p is a prime. It was shown in L. Xu et al. (1999) that finding a perfect 1-factorization of Kn can be reduced to a problem in coding, i.e. to constructing an MDS, lowest density array code of length n. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the Kp+1 family of perfect 1-factorizations from the K 2p family, by applying the reduction mentioned above. Namely, techniques from coding theory are used to prove a new result in graph theory","PeriodicalId":92224,"journal":{"name":"International Symposium on Information Theory and its Applications. International Symposium on Information Theory and its Applications","volume":"26 1","pages":"2799-2803"},"PeriodicalIF":0.0000,"publicationDate":"2006-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Information Theory and its Applications. International Symposium on Information Theory and its Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ISIT.2006.261572","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
Abstract
The existence of a perfect 1-factorization of the complete graph K n, for arbitrary n, is a 40-year old open problem in graph theory. Two infinite families of perfect 1-factorizations are known for K2p and Kp+1, where p is a prime. It was shown in L. Xu et al. (1999) that finding a perfect 1-factorization of Kn can be reduced to a problem in coding, i.e. to constructing an MDS, lowest density array code of length n. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the Kp+1 family of perfect 1-factorizations from the K 2p family, by applying the reduction mentioned above. Namely, techniques from coding theory are used to prove a new result in graph theory
对于任意n的完全图K n的完全1分解的存在性，是图论中一个有40年历史的开放问题。两个无限族的完全1因子分解已知为K2p和Kp+1，其中p是素数。L. Xu et al.(1999)表明，寻找Kn的完美1因子分解可以归结为一个编码问题，即构造一个长度为n的最小密度阵列码MDS。本文介绍了一种缩短任意阵列码的新方法。然后用它从k2p族推导出Kp+1族的完全1因子分解，通过应用上面提到的简化。也就是说，利用编码理论中的技术来证明图论中的一个新结果
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
