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The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, K\H{o}v\'ari S\'os and Tur\'an proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d

A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this note we establish a (modest) strengthening of old results of Hansel and of Katona and Szemer\'edi, by showing that the capacity of any bipartite covering of a graph on $n$ vertices in which the maximum size of an independent set containing vertex number $i$ is $\alpha_i$, is at least $\sum_i \log_2 (n/\alpha_i).$ We also obtain slightly improved bounds for a recent result of Kim and Lee about the minimum possible capacity of a bipartite covering of complete multigraphs.

We investigate the twin-width of the Erd\H{o}s-R\'enyi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $\Theta(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$., Comment: 37 pages, 3 figures

We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.

The Zarankiewicz function gives, for a chosen matrix and minor size, the maximum number of ones in a binary matrix not containing an all-one minor. Tables of this function for small arguments have been compiled, but errors are known in them. We both correct the errors and extend these tables in the case of square minors by expressing the problem of finding the value at a specific point as a series of Boolean satisfiability problems, exploiting permutation symmetries for a significant reduction in the work needed. When the ambient matrix is also square we also give all non-isomorphic examples of matrices attaining the maximum, up to the aforementioned symmetries; it is found that most maximal matrices have some form of symmetry., Comment: 24 pages, 2 figures. Added correctness proof of partition-generating algorithm and a new subsection on coverings

Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$., Comment: 22 pages, 1 figure

We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. "A semi-algebraic version of Zarankiewicz's problem"] on the semialgebraic planar Zarankiewicz problem to arbitrary $o$-minimal structures, in particular obtaining an $o$-minimal generalization of the Szemer\'edi-Trotter theorem.

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. "A semi-algebraic version of Zarankiewicz's problem"] on the semialgebraic planar Zarankiewicz problem to arbitrary $o$-minimal structures, in particular obtaining an $o$-minimal generalization of the Szemer\'edi-Trotter theorem.

We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k

Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{\lfloor n^2/4\rfloor}$ for $n \geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n \geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{\lfloor n^2/4\rfloor}$ such orientations.

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. K\'aszonyi and Tuza, in 1986, gave a general upper bound on the saturation number of a graph $H$, but a nontrivial lower bound has remained elusive. In this paper we give a general lower bound on $\mathrm{sat}(H,n)$ and prove that it is asymptotically sharp (up to an additive constant) on a large class of graphs. This class includes all threshold graphs and many graphs for which the saturation number was previously determined exactly. Our work thus gives an asymptotic common generalization of several earlier results. The class also includes disjoint unions of cliques, allowing us to address an open problem of Faudree, Ferrara, Gould, and Jacobson., Comment: 10 pages, 2 figures. Fixed a few minor typos from the first version

We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate $H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an $H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge \left(\frac{1}{2} + d^{-1+\Omega(r^{-1})}\right)m$, generalizing a recent work of Carlson, Kolla, and Trevisan. We also give bounds on $f(m,d,H)$ when $H$ is a cycle, odd wheel, or a complete bipartite graph with at most 4 vertices on one side. We also show stronger bounds on $f(m,d,K_r)$ assuming a conjecture of Alon, Bollabas, Krivelevich, and Sudakov (2003). We conjecture that $f(m,d,K_r)= \left( \frac{1}{2} + \Theta_r(d^{-1/2}) \right)m$ for every $r\ge 3$, and show that this conjecture implies the ABKS conjecture., Comment: This paper has been superseded by arXiv:1810.10044

We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lov\'asz, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-2466] and [M. Malliaris, S. Shelah, "Regularity lemmas for stable graphs", Transactions of the American Mathematical Society, 366.3, 2014, 1551-1585]. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples.

We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lov\'asz, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-2466] and [M. Malliaris, S. Shelah, "Regularity lemmas for stable graphs", Transactions of the American Mathematical Society, 366.3, 2014, 1551-1585]. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples.

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph which was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced. In this paper we study the F-index of four operations on graphs which were introduced by Eliasi and Taeri [M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, \textit{Discrete Appl. Math.}\textbf{157}(2009) 794--803.]., Comment: 13 pages

In this note, we study a dynamic vertex coloring for a graph $G$. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a \emph{zero forcing set} if by iterating this process, all of the vertices in $G$ become black. The \emph{zero forcing number} of $G$ is the minimum cardinality of a zero forcing set in $G$, and is denoted by $Z(G)$. Davila and Kenter have conjectured in 2015 that $Z(G)\geq (g-3)(\delta-2)+\delta$ where $g$ and $\delta$ denote the girth and the minimum degree of $G$, respectively. This conjecture has been proven for graphs with girth $g \leq 10$. In this note, we present a proof for $g \geq 5$, $\delta \geq 2$, thereby settling the conjecture.

The Narumi-Katayama index of a graph was introduced in 1984 for representing the carbon skeleton of a saturated hydrocarbons and is defined as the product of degrees of all the vertices of the graph. In this paper, we examine the Narumi-Katayama index of different total transformation graphs., Comment: 15 Pages. arXiv admin note: text overlap with arXiv:1606.05989

The F-index of a graph $G$ is the sum of the cubes of the degrees of the vertices of $G$. In this paper, explicit expressions for the F-index of different transformation graphs of type ${{G}^{xyz}}$ with $x, y, z\in \{-, + \}$ are obtained. F-index for semitotal point graph and semitotal line graph are also obtained here., Comment: 10 pages

Este artículo presenta un análisis de las medidas de irregularidad de grafos que se encuentran en la literatura. Desde su discusión dos puntos importantes aparecen: primero, la ausencia de relación entre todos ellos, sino una sola excepción - con las estructuras de los grafos correspondientes y, además, sus valores extremales conocidos corresponden a grafos que tienen secuencias de grados con pocos valores diferentes. Se proponen dos nuevas medidas, tratando de cumplir con estos puntos. Sus valores se calculan para grafos extremales asociados con otras medidas y para grafos antiregulares. Por último, se determinan los gráficos de quartiles o boxplots de todas estas medidas, para algunos conjuntos de grafos de la literatura y para cuatro conjuntos donde las secuencias ordenadas de grados son constantes. Todas las medidas estudiadas tienen complejidad polinómica., This paper presents some measures of graph irregularity found in the literature. From their discussion two important points appear: first, the absence of relationship between all of them, but a single exception, with the structures of the corresponding graphs and, moreover, their known extremal values correspond to graphs having degree sequences with few different values. Two new measures are proposed, seeking to meet these points. Their values are calculated for extremal graphs associated with other measures and for antiregular graphs. Finally, we calculate the boxplots of all these measures for some sets of graphs taken from the literature and also for four sets where the ordered degree sequences are constant. All measures involved have polynomial complexity.

The total eccentricity index of a graph \(G\) is the sum of eccentricities of all the vertices of \(G\). In this paper, we first derive some sharp upper and lower bounds of total eccentricity index of different subdivision graphs and then determine some explicit expression of the total eccentricity index of the double graph, extended double cover graph and some generalized thorn graphs.

The total eccentricity index of a graph \(G\) is the sum of eccentricities of all the vertices of \(G\). In this paper, we first derive some sharp upper and lower bounds of total eccentricity index of different subdivision graphs and then determine some explicit expression of the total eccentricity index of the double graph, extended double cover graph and some generalized thorn graphs.

Este artículo presenta un análisis de las medidas de irregularidad de grafos que se encuentran en la literatura. Desde su discusión dos puntos importantes aparecen: primero, la ausencia de relación entre todos ellos, sino una sola excepción - con las estructuras de los grafos correspondientes y, además, sus valores extremales conocidos corresponden a grafos que tienen secuencias de grados con pocos valores diferentes. Se proponen dos nuevas medidas, tratando de cumplir con estos puntos. Sus valores se calculan para grafos extremales asociados con otras medidas y para grafos antiregulares. Por último, se determinan los gráficos de quartiles o boxplots de todas estas medidas, para algunos conjuntos de grafos de la literatura y para cuatro conjuntos donde las secuencias ordenadas de grados son constantes. Todas las medidas estudiadas tienen complejidad polinómica., This paper presents some measures of graph irregularity found in the literature. From their discussion two important points appear: first, the absence of relationship between all of them, but a single exception, with the structures of the corresponding graphs and, moreover, their known extremal values correspond to graphs having degree sequences with few different values. Two new measures are proposed, seeking to meet these points. Their values are calculated for extremal graphs associated with other measures and for antiregular graphs. Finally, we calculate the boxplots of all these measures for some sets of graphs taken from the literature and also for four sets where the ordered degree sequences are constant. All measures involved have polynomial complexity.

We examine several types of visibility graphs in which sightlines can pass through $k$ objects. For $k \geq 1$ we bound the maximum thickness of semi-bar $k$-visibility graphs between $\lceil \frac{2}{3} (k + 1) \rceil$ and $2k$. In addition we show that the maximum number of edges in arc and circle $k$-visibility graphs on $n$ vertices is at most $(k+1)(3n-k-2)$ for $n > 4k+4$ and ${n \choose 2}$ for $n \leq 4k+4$, while the maximum chromatic number is at most $6k+6$. In semi-arc $k$-visibility graphs on $n$ vertices, we show that the maximum number of edges is ${n \choose 2}$ for $n \leq 3k+3$ and at most $(k+1)(2n-\frac{k+2}{2})$ for $n > 3k+3$, while the maximum chromatic number is at most $4k+4$., Comment: 17 pages, 6 figures

The classical paradox of social choice theory asserts that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on N vertices (one per candidate), placing an edge from U to V if U would beat V in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a complete binary tree whose leaves are labeled by the candidates, and evaluate it by running pairwise elections between the pairs of leaves, sending the winners to successive rounds of pairwise elections which ultimately terminate with a single winner. This structure is called a voting tree. Much research has investigated which functions on tournaments are computable in this way. Fischer, Procaccia, and Samorodnitsky quantitatively studied the computability of the Copeland rule, which returns a vertex of maximum out-degree in the given tournament. Perhaps surprisingly, the best previously known voting tree could only guarantee a returned out-degree of at least log_2 N, despite the fact that every tournament has a vertex of degree at least (N-1)/2. In this paper, we present three constructions, the first of which substantially improves this guarantee to \Theta(sqrt{N}). The other two demonstrate the richness of the voting tree universe, with a tree that resists manipulation, and a tree which implements arithmetic modulo three., Comment: 13 pages

We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number $n$ of maximal independent sets for a class of natural numbers that can be represented as concatenation of periodic binary words., Comment: 1 figure, 8 pages

We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and, more generally, to Schubert varieties in Grassmannians. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a closed family. We propose a generalization of this to what is called a subclose family. A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined., Comment: 11 pages

We prove that there is no cubic graph with diameter 4 on 40 vertices. This implies that the maximal number of vertices of a (3,4)-graph is 38. © 2000 Elsevier Science B.V., SCOPUS: ar.j, info:eu-repo/semantics/published

We prove that there is no cubic graph with diameter 4 on 40 vertices. This implies that the maximal number of vertices of a (3,4)-graph is 38. © 2000 Elsevier Science B.V., SCOPUS: ar.j, info:eu-repo/semantics/published