{"title":"Large Cycles in Graphs around Bondy’s and Jung’s Conjectures – Modifications, Sharpness, and Perspectives","authors":"Zh. G. Nikoghosyan","doi":"10.1134/s1054661824010140","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In 1980, Bondy conjectured a common generalization (depending on λ) of some well-known degree-sum conditions for a graph ensuring the existence of Hamilton cycles for λ = 1 (Ore, 1960) and dominating cycles for λ = 2 (Bondy, 1980) as special cases. The reverse (long-cycle) version of Bondy’s conjecture was proposed in 2001 due to Jung. The importance of these two conjectures in the field is motivated by the fact that they (as starting points) give rise to all (with few exceptions) further developments through various additional extensions and limitations. In this paper, we briefly outline all known notable achievements towards solving the problem: (i) confirmation (by the author) of Bondy’s and Jung’s conjectures for some versions that are very close to the original versions; and (ii) significant improvements (by the author) of results (i), inspiring a number of improved versions of original conjectures of Bondy and Jung. Next we derive a number of modifications from improvements in (ii), which are also very close to the original versions, but do not follow directly from the Bondy’s and Young’s conjectures. Finally, all results (both old and new) are shown to be best possible in a sense based on three types of sharpness, indicating the intervals in 0 < λ < δ + 1 where the result is sharp and the intervals where the result can be further improved, where δ denotes the minimum degree.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":"17 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1054661824010140","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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Abstract
In 1980, Bondy conjectured a common generalization (depending on λ) of some well-known degree-sum conditions for a graph ensuring the existence of Hamilton cycles for λ = 1 (Ore, 1960) and dominating cycles for λ = 2 (Bondy, 1980) as special cases. The reverse (long-cycle) version of Bondy’s conjecture was proposed in 2001 due to Jung. The importance of these two conjectures in the field is motivated by the fact that they (as starting points) give rise to all (with few exceptions) further developments through various additional extensions and limitations. In this paper, we briefly outline all known notable achievements towards solving the problem: (i) confirmation (by the author) of Bondy’s and Jung’s conjectures for some versions that are very close to the original versions; and (ii) significant improvements (by the author) of results (i), inspiring a number of improved versions of original conjectures of Bondy and Jung. Next we derive a number of modifications from improvements in (ii), which are also very close to the original versions, but do not follow directly from the Bondy’s and Young’s conjectures. Finally, all results (both old and new) are shown to be best possible in a sense based on three types of sharpness, indicating the intervals in 0 < λ < δ + 1 where the result is sharp and the intervals where the result can be further improved, where δ denotes the minimum degree.
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The purpose of the journal is to publish high-quality peer-reviewed scientific and technical materials that present the results of fundamental and applied scientific research in the field of image processing, recognition, analysis and understanding, pattern recognition, artificial intelligence, and related fields of theoretical and applied computer science and applied mathematics. The policy of the journal provides for the rapid publication of original scientific articles, analytical reviews, articles of the world''s leading scientists and specialists on the subject of the journal solicited by the editorial board, special thematic issues, proceedings of the world''s leading scientific conferences and seminars, as well as short reports containing new results of fundamental and applied research in the field of mathematical theory and methodology of image analysis, mathematical theory and methodology of image recognition, and mathematical foundations and methodology of artificial intelligence. The journal also publishes articles on the use of the apparatus and methods of the mathematical theory of image analysis and the mathematical theory of image recognition for the development of new information technologies and their supporting software and algorithmic complexes and systems for solving complex and particularly important applied problems. The main scientific areas are the mathematical theory of image analysis and the mathematical theory of pattern recognition. The journal also embraces the problems of analyzing and evaluating poorly formalized, poorly structured, incomplete, contradictory and noisy information, including artificial intelligence, bioinformatics, medical informatics, data mining, big data analysis, machine vision, data representation and modeling, data and knowledge extraction from images, machine learning, forecasting, machine graphics, databases, knowledge bases, medical and technical diagnostics, neural networks, specialized software, specialized computational architectures for information analysis and evaluation, linguistic, psychological, psychophysical, and physiological aspects of image analysis and pattern recognition, applied problems, and related problems. Articles can be submitted either in English or Russian. The English language is preferable. Pattern Recognition and Image Analysis is a hybrid journal that publishes mostly subscription articles that are free of charge for the authors, but also accepts Open Access articles with article processing charges. The journal is one of the top 10 global periodicals on image analysis and pattern recognition and is the only publication on this topic in the Russian Federation, Central and Eastern Europe.
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