Clique Partitions and Clique Coverings

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Revisiting path-type covering and partitioning problems

This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts

Algorithmic approaches to problems in probabilistic combinatorics

The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method: 1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse. 2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n). 3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory. 4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D

Workshop Notes of the Sixth International Workshop "What can FCA do for Artificial Intelligence?"

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