Given two non-negative integers $n$ and $s$, define $m(n,s)$ to be the maximal number such that in every hypergraph $\mathcal{H}$ on $n$ vertices and with at most $ m(n,s)$ edges there is a vertex $x$ such that $|\mathcal{H}_x|\geq | E(\mathcal{H})| -s$, where $\mathcal{H}_x=\{H\setminus\{x\}:H\in E(\mathcal{H})\}$. This problem has been posed by Füredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of $s$, Frankl determined $m(n,2^{d-1}-1)$ for all $d\in\mathbb{N}$ with $d\mid n$. Subsequently, the goal became to determine $m(n,2^{d-1}-c)$ for larger $c$. Frankl and Watanabe determined $m(n,2^{d-1}-c)$ for $c\in\{0,2\}$. Other general results were not known so far.
Our main result sheds light on what happens further away from powers of two: We prove that $m(n,2^{d-1}-c)=\frac{n}{d}(2^d-c)$ for $d\geq 4c$ and $d\mid n$ and give an example showing that this equality does not hold for $c=d$. The other line of research on this problem is to determine $m(n,s)$ for small values of $s$. In this line, our second result determines $m(n,2^{d-1}-c)$ for $c\in\{3,4\}$. This solves more instances of the problem for small $s$ and in particular solves a conjecture by Frankl and Watanabe.
The Wiener polarity index $W_p$, one of the most studied molecular structure descriptors, was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index $W_p$ for chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3. A vertex of a chemical tree with the degree at least 3 is called a branching vertex. A segment of a chemical tree $T$ is a path-subtree $S$ whose terminal vertices have degrees different from 2 in $T$ and every internal vertex (if exists) of $S$ has degree 2 in $T$. In this pap…
In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our more significant contributions on λ-codes and η-codes concerning concerning bounds, extremal values and realization theorems.
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.
The famous lower bound $α(G)\geq \sum_{u\in V(G)}\frac{1}{d_G(u)+1}$ on the independence number $α(G)$ of a graph $G$ due to Caro and Wei is known to be tight if and only if the components of $G$ are cliques, and has been generalized several times in the context of large degenerate subsets and small dynamic monopolies. We characterize the extremal graphs for a generalization due to Ackerman, Ben-Zwi, and Wolfovitz. Furthermore, we give a simple proof of a related bound concerning partial incentives due to Cordasco, Gargano, Rescigno, and Vaccaro, and also characterize the corresponding extrema…
The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptio…
In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the sa…
In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is a polarity graph of a projective plane of order $q$ that has an oval, then $G$ contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O ( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known lower bounds on the Turán number $\mathrm{ex}(n, C_4)$ for certain values of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.
In this paper, we determine the maximum number of nonzero entries in 0-1 matrices of order $n$ with zero trace whose squares are 0-1 matrices when $n\ge 8$. The extremal matrices attaining this maximum number are also characterized.
In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the structured component, is used in many applications to extremal problems. An analogous decomposition theorem exists for functions over $\mathbb{F}_p^n$. Specializing to $p=2$, we obtain a statement about the indicator functions of simple binary matroids. In this paper we extend previous r…
In this paper we introduce a unifying approach to the generalized Turán problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least $m$ copies of $F$ as a subgraph. We present a survey, discuss previously known results and obtain several new ones focusing mainly on proof methods, extremal structure and phase transition phenomena. Finally we point out some relation with extremal questions concerning hypergraphs, particularly Berge-type results…
In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. In this paper some of this research is extended to more general matrix norms, like the Schatten p-norms and the Ky Fan k-norms. Whenever possible the results are given both for graphs and general matrices. In various contexts a puzzling fact was observed: the Schatten p-norms are widely different for 1<=p<2 and for p>=2.
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
An edge coloring $c$ of a graph $G$ is a royal $k$-edge coloring of $G$ if the edges of $G$ are assigned nonempty subsets of the set $\{1, 2, \ldots, k\}$ in such a way that the vertex coloring obtained by assigning the union of the colors of the incident edges of each vertex is a proper vertex coloring. If the vertex coloring is vertex-distinguishing, then $c$ is a strong royal $k$-edge coloring. The minimum positive integer $k$ for which $G$ has a strong royal $k$-edge coloring is the strong royal index of $G$. It has been conjectured that if $G$ is a connected graph of order $n\ge 4$ where …
Classical questions in extremal graph theory concern the asymptotics of $\operatorname{ex}(G, \mathcal{H})$ where $\mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard' increasing sequence of host graphs $(G_1, G_2, \dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $\operatorname{ex}(G,\mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $\mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context,…
For positive integers $w$ and $k$, two vectors $A$ and $B$ from $\mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\geq k$ and $B[j]-A[j]\geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $\mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natur…
Technion, Israel Institute of Technology（以色列理工学院）；Microsoft Research（微软研究院）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；University of Santa Cruz（圣克鲁斯大学）
It is a long-standing open problem whether there always exists a compression scheme whose size is of the order of the Vapnik-Chervonienkis (VC) dimension $d$. Recently compression schemes of size exponential in $d$ have been found for any concept class of VC dimension $d$. Previously, compression schemes of size $d$ have been given for maximum classes, which are special concept classes whose size equals an upper bound due to Sauer-Shelah. We consider a generalization of maximum classes called extremal classes. Their definition is based on a powerful generalization of the Sauer-Shelah bound cal…
Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $σ:V(X)\to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $σ(a)$ and $σ(b)$ are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected; we address this problem from two different perspectives. First, we address the case of "typica…
RLGT: A reinforcement learning framework for extremal graph theory
Faculty of Electronic Engineering, University of Niš（尼什大学电子工程学院）；Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska（普里摩尔卡大学数学、自然科学与信息科技学院）；Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）
