Hamilton cycles in graphs and hypergraphs: an extremal perspective
نویسنده
چکیده
As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches. Mathematics Subject Classification (2010). Primary 05C45; Secondary 05C35; 05C65; 05C20.
منابع مشابه
Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs
This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....
متن کاملOn Spanning Structures in Random Hypergraphs
In this note we adapt a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125–148] from random graphs to random r-uniform hypergaphs. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs.
متن کاملOn the Number of Hamilton Cycles in Bounded Degree Graphs
The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs – deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276) for the maximum number of Hamilton cycles in 3-regul...
متن کاملExtremal hypergraph theory and algorithmic regularity lemma for sparse graphs
Once invented as an auxiliary lemma for Szemerédi’s Theorem [106] the regularity lemma [105] has become one of the most powerful tools in graph theory in the last three decades which has been widely applied in several fields of mathematics and theoretical computer science. Roughly speaking the lemma asserts that dense graphs can be approximated by a constant number of bipartite quasi-random gra...
متن کاملCounting and packing Hamilton cycles in dense graphs and oriented graphs
We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c > 3/8 contains (cn/e)(1 + o(1)) directed Hamilton cycles. This is an extension of a result of Cuckler, who settle...
متن کامل