We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers k ⩾ 1 , r ⩾ 0, and ℓ ⩾ ( r + 1 ) r, and for any α > k k + 1 we show that adding O ( n 2 − 2 ∕ ℓ ) random edges to an n‐vertex graph G with minimum degree at least α n yields, with probability close to one, the existence of the ( k ℓ + r )‐th power of a Hamiltonian cycle. In particular, for r = 1 and ℓ = 2 this implies that adding O ( n ) random edges to such a graph G already ensures the ( 2 k + 1 )‐st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of k , ℓ, and r we can show that our result is asymptotically optimal.
Abstract. Turán's theorem is a cornerstone of extremal graph theory. It asserts that for any integer r ě 2 every graph on n vertices with more than r´2 2pr´1q¨n 2 edges contains a clique of size r, i.e., r mutually adjacent vertices. The corresponding extremal graphs are balanced pr´1q-partite graphs.The question as to how many such r-cliques appear at least in any n-vertex graph with γn 2 edges has been intensively studied in the literature. In particular, Lovász andSimonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by the complete multipartite graph with γn 2 edges in which all but one vertex class is of the same size while the remaining one may be smaller.Their conjecture was recently resolved for r " 3 by Razborov and for r " 4 by Nikiforov. In this article, we prove the conjecture for all values of r. §1. Introduction
In 1961, Erdős, Ginzburg and Ziv proved a remarkable theorem stating that each set of 2n − 1 integers contains a subset of size n, the sum of whose elements is divisible by n. We will prove a similar result for pairs of integers, i.e. planar latticepoints, usually referred to as Kemnitz' conjecture.
Extremal problems for 3-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a 3-uniform hypergraph H " pV, Eq is weakly pd, ηq-quasirandom if for any subset U Ď V the number of hyperedges of H contained in U is in the interval d`| U | 3˘˘η |V | 3 . We show that for any ε ą 0 there exists η ą 0 such that every sufficiently large weakly p1{4`ε, ηqquasirandom hypergraph contains four vertices spanning at least three hyperedges. This was conjectured by Erdős and Sós and it is known that the density 1{4 is best possible.Recently, a computer assisted proof of this result based on the flag-algebra method was established by Glebov, Kráľ, and Volec. In contrast to their work our proof presented here is based on the regularity method of hypergraphs and requires no heavy computations. In addition we obtain an ordered version of this result. The method of our proof allows us to study extremal problems of this type in a more systematic way and we discuss a few extensions and open problems here. n 2w hich describes the maximum density of large F -free graphs. The systematic study of these extremal parameters was initiated by Turán [34], who determined expn, K k q for complete graphs K k . Recalling that the chromatic number χpF q of a graph F is the minimum number 2010 Mathematics Subject Classification. 05C35 (primary), 05C65, 05C80 (secondary).
Abstract. We study the notion of formal duality introduced by Cohn, Kumar, and Schürmann in their computational study of energy-minimizing particle configurations in Euclidean space. In particular, using the Poisson summation formula we reformulate formal duality as a combinatorial phenomenon in finite abelian groups. We give new examples related to Gauss sums and make some progress towards classifying formally dual configurations.
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