Related papers: Elusive extremal graphs
For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erd\H os--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov\'asz and Simonovits from…
In this paper, we use various versions of Lov\'asz extension to systematically derive continuous formulations of problems from discrete mathematics. This will take place in the following context: (1) For combinatorial optimization problems…
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit…
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We…
This paper aims to maximize algebraic connectivity of networks via topology design under the presence of constraints and an adversary. We are concerned with three problems. First, we formulate the concave maximization topology design…
In the 1960s, Erd\H{o}s and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by…
A fundamental problem in pattern avoidance is describing the asymptotic behavior of the extremal function and its generalizations. We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite…
This article connects the theory of extremal doubly stochastic measures to the geometry and topology of optimal transportation. We begin by reviewing an old question (# 111) of Birkhoff in probability and statistics [4], which is to give a…
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
This paper consists of two halves. In the first half of the paper, we consider real-valued functions $f$ whose domain is the vertex set of a graph $G$ and that are Lipschitz with respect to the graph distance. By placing a uniform…
Lov\'{a}sz and Cherkassky discovered independently that, if $G$ is a finite graph and $T\subseteq V(G)$ such that the degree $d_G(v)$ is even for every vertex $v\in V(G)\setminus T$, then the maximum number of edge-disjoint paths which are…
Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability…
We examine the Maximum Independent Set Problem in an undirected graph. The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs. The problem is…
The theory of limits of dense graph sequences was initiated by Lovasz and Szegedy. We give a possible generalization of this theory to multigraphs. Our proofs are based on the correspondence between dense graph limits and countable,…
We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex…
We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory…
We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.
Define f(n,p) to be the maximum number of edges in a graph on n vertices with p perfect matchings. Dudek and Schmitt proved there exist constants n_p and c_p so that for even n >= n_p, f(n,p) = (n^2)/4+c_p. A graph is p-extremal if it has p…
For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…