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The planar Turán numberexP(n,H) of a graph H is the maximum number of edges in an n‐vertex planar graph without H as a subgraph. Let Ck denote the cycle of length k. The planar Turán number exP(n,Ck) is known for k≤7. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant D so that exP(n,Ck)≤3n−6−Dn/klog2 3 for all k≥4 and n≥klog2 3. When k≥11 this bound is tight up to the constant D and proves a conjecture of Cranston, Lidický, Liu, and Shantanam. [ABSTRACT FROM AUTHOR]

Clique counts are crucial in applications like detecting communities in social networks and recurring patterns in bioinformatics. Counting k-cliques—a fully connected subgraph with k nodes, where each node has a direct, mutual, and symmetric relationship with every other node—becomes computationally challenging for larger k due to combinatorial explosion, especially in large, dense graphs. Existing exact methods have difficulties beyond k = 10, especially on large datasets, while sampling-based approaches often involve trade-offs in terms of accuracy, resource utilization, and efficiency. This difficulty becomes more pronounced in dense graphs as the number of potential k-cliques grows exponentially. We present Boundary-driven approximations of k-cliques (BDAC), a novel algorithm that approximates k-clique counts without using recursive procedures or sampling methods. BDAC offers both lower and upper bounds for k-cliques at local (per-vertex) and global levels, making it ideal for large, dense graphs. Unlike other approaches, BDAC's complexity remains unaffected by the value of k. We demonstrate its effectiveness by comparing it with leading algorithms across various datasets, focusing on k values ranging from 8 to 50. [ABSTRACT FROM AUTHOR]

Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs in Chvátal and Hanson (J Combin Theory Ser B 20:128–138, 1976) and Balachandran and Khare (Discrete Math 309:4176–4180, 2009). It follows from the structure of those extremal graphs that deciding whether this maximum number decreases or not when restricted to claw-free graphs, to C 4 -free graphs or to triangle-free graphs are separately interesting research questions. The first two cases being already settled in Dibek et al. (Discrete Math 340:927–934, 2017) and Blair et al. (Latin American symposium on theoretical informatics, 2020), in this paper we focus on triangle-free graphs. We show that unlike most cases for claw-free graphs and C 4 -free graphs, forbidding triangles from extremal graphs causes a strict decrease in the number of edges and adds to the hardness of the problem. We provide a formula giving the maximum number of edges in a triangle-free graph with degree at most d and matching number at most m for all cases where d ≥ m , and for the cases where d < m with either d ≤ 6 or Z (d) ≤ m < 2 d where Z(d) is a function of d which is roughly 5d/4. We also provide an integer programming formulation for the remaining cases and as a result of further discussion on this formulation, we conjecture that our formula giving the size of triangle-free extremal graphs is also valid for these open cases. [ABSTRACT FROM AUTHOR]

We consider special distance graphs and estimate the number of edges in their subgraphs. The estimates obtained improve some known results. [ABSTRACT FROM AUTHOR]

Clique counts are crucial in applications like detecting communities in social networks and recurring patterns in bioinformatics. Counting k-cliques—a fully connected subgraph with k nodes, where each node has a direct, mutual, and symmetric relationship with every other node—becomes computationally challenging for larger k due to combinatorial explosion, especially in large, dense graphs. Existing exact methods have difficulties beyond k = 10, especially on large datasets, while sampling-based approaches often involve trade-offs in terms of accuracy, resource utilization, and efficiency. This difficulty becomes more pronounced in dense graphs as the number of potential k-cliques grows exponentially. We present Boundary-driven approximations of k-cliques (BDAC), a novel algorithm that approximates k-clique counts without using recursive procedures or sampling methods. BDAC offers both lower and upper bounds for k-cliques at local (per-vertex) and global levels, making it ideal for large, dense graphs. Unlike other approaches, BDAC’s complexity remains unaffected by the value of k. We demonstrate its effectiveness by comparing it with leading algorithms across various datasets, focusing on k values ranging from 8 to 50.

Ramsey's theory (RAM) from combinatorics and network theory goes looking for regularities and repeated patterns inside structures equipped with nodes and edges. RAM represents the outcome of a dual methodological commitment: by one side a top-down approach evaluates the possible arrangement of specific subgraphs when the number of graph's vertices is already known, by another side a bottom-up approach calculates the possible number of graph's vertices when the arrangement of specific subgraphs is already known. Since natural neural networks are often represented in terms of graphs, we propose to use RAM for the analytical and computational assessment of the human brain connectome, i.e., the distinctive anatomical/functional structure provided with vertices and edges at different coarse-grained scales. We retrospectively examined the literature regarding graph theory in neuroscience, looking for hints that might be related to the RAM framework. At first, we establish that the peculiar network required by RAM, i.e., a graph characterized by every vertex connected with all the others, does exist inside the human brain connectome. Then, we argue that the RAM approach to nervous networks might be able to trace unexpected functional interactions and unexplored motifs shared between different cortical and subcortical subareas. Furthermore, we describe how remarkable RAM outcomes, such as the Ramsey's theorem and the Ramsey's number, might contribute to quantify novel properties of neuronal networks and uncover still unknown anatomical connexions. [ABSTRACT FROM AUTHOR]

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called edge-ordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edge-order but the same underlying graph may appear with a different edge-order. We prove an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs—we identify the relevant parameter for the Turán number of an edge-ordered graph and call it the order chromatic number. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in discrete geometry. [ABSTRACT FROM AUTHOR]

The celebrated Motzkin–Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Turán's theorem in graph theory, but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization. Clique relaxations, such as s-defective clique and s-plex, emerged as attractive, more practical alternatives to cliques in network-based cluster detection models arising in numerous applications. This paper establishes continuous cubic formulations for the maximum s-defective clique problem and the maximum s-plex problem by generalizing the Motzkin–Straus formulation to the corresponding clique relaxations. The formulations are used to extend Turán's theorem and other known lower bounds on the clique number to the considered clique relaxations. Results of preliminary numerical experiments with the CONOPT solver demonstrate that the proposed formulations can be used to quickly compute high-quality solutions for the maximum s-defective clique problem and the maximum s-plex problem. The proposed formulations can also be used to generate interesting test instances for global optimization solvers. [ABSTRACT FROM AUTHOR]

One of the most important Turán's theorems establishes an inequality between the maximum clique and the number of edges of a graph. Since 1941, this result has received much attention and many of the different proofs involve induction and a probability distribution. In this paper we detail finite procedures that gives a proof for the Turán's Theorem. Among other things, we give a generalization of this result. Also we apply this results to a Nikiforov's inequality between the spectral radius and the maximum clique of a graph. [ABSTRACT FROM AUTHOR]

In Henning and Jafari Rad (Graphs Combin, 37: 325–336, 2021), several new probabilistic upper bounds are given on the k-tuple domination number, k-domination number, Roman domination number, and Roman k-domination number of a graph using the well-known Brooks’ Theorem for vertex coloring, improving all of previous given bounds for the above domination variants. In this paper, we use the well-known Turán’s Theorem, and give a slight improvement of all above given bounds. [ABSTRACT FROM AUTHOR]

A conjecture of Chung and Graham states that every K4‐free graph on n vertices contains a vertex set of size ⌊n∕2⌋ that spans at most n2∕18 edges. We make the first step toward this conjecture by showing that it holds for all regular graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with m edges and let ρ be the largest eigenvalue of its adjacency matrix. It is shown that

Given a measurable set A ⊂ R d we consider the large-distance graph G A , on the ground set A, in which each pair of points from A whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the 2d-dimensional Lebesgue measure of the edge set as well as the d-dimensional Lebesgue measure of the vertex set of a large-distance graph in the d-dimensional Euclidean space that contains no copies of a complete graph on k vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latter as a "graph-theoretic" analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest. [ABSTRACT FROM AUTHOR]

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H‐free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2‐balanced T. Our results in the case when m2(H) > m2(T) are a natural generalization of the Erdős‐Stone theorem for G(n,p), proved several years ago by Conlon‐Gowers and Schacht; the case T = Km was previously resolved by Alon, Kostochka, and Shikhelman. The case when m2(H) ≤ m2(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán‐type problems that we are unable to solve in full generality. [ABSTRACT FROM AUTHOR]

A subset $S$ of vertices of a graph $G$ is an \textit{independent set} if no pair of vertices of $S$ are adjacent. The \textit{independence number}, $\alpha(G)$ of $G$, is the maximum cardinality of an independent set of $G$. In this note, we present an improvement of the weak Tur$\acute{a}$n's theorem.

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n with vertex set V(G). For positive integers a and b, a subset S ⊆ V (G) is an (a, b)-dominating set if every vertex v ∈ S is adjacent to at least a vertices inside S and every vertex v ∈ V \ S is adjacent to at least b vertices inside S. The minimum cardinality of an (a, b)-dominating set for G is called the (a, b)-domination number of G and is denoted by γ a , b (G) . There are various results on upper bounds for γ a , b (G) when G is a regular graph or a and b are small numbers. In the first part of this paper, for a given graph G with the minimum degree of at least max { a , b } , we define a new graph G ′ associated to G and show that the independence number of this graph is related to γ a , b (G) . In the next part, using Lovász local lemma, we give a randomized approach to improve previous results on the upper bounds for γ a , b (G) in some special cases. [ABSTRACT FROM AUTHOR]

We obtain new estimates for the number of edges in induced subgraphs of some distance graph. [ABSTRACT FROM AUTHOR]

An extremal result about vertex covers, attributed by Hajnal to Erdős and Gallai, is applied to prove the following: If n, k, and t are integers satisfying n ≥ k ≥ t ≥ 3 and k ≤ 2t - 2, and G is a graph with the minimum number of edges among graphs on n vertices with the property that every induced subgraph on k vertices contains a complete subgraph on t vertices, then every component of G is complete.

We consider the comparison complexity of finding modes (the most frequently occurring elements) in a list of elements that are not necessarily from a totally ordered set. Here, the relation between elements is determined by equality comparisons whose outcome is = when the two elements being compared are equal, and ≠ otherwise. The problem generalizes the classical majority problem studied in this model (using equalities). We show that n 2 / 2 m − n / 2 comparisons are necessary and n 2 / m + n comparisons are sufficient to find an element that appears at least m times. This is in sharp contrast to the bound of Θ ( n log ⁡ ( n / m ) ) bound in the model where comparisons are < , = , > or ≤ , > . We give three algorithms for finding mode, including one that is a generalization of a classical majority finding algorithm due to Fischer and Salzberg (1982) [9] . We also discuss upper and lower bounds for sorting (i.e., finding the frequency of every element) and for finding the least frequent element. Sorting problem (under the equality comparisons) also known as equivalence class sorting , has applications in several scenarios where the total order of elements is either not possible or can not be revealed for security reasons. [ABSTRACT FROM AUTHOR]

A subset S of vertices of a graph G is an independent set if no pair of vertices of S are adjacent. The independence number, α(G) of G, is the maximum cardinality of an independent set of G. In this note, we present an improvement of the weak Turán's theorem. [ABSTRACT FROM AUTHOR]

Let F be a graph that contains an edge whose deletion reduces its chromatic number. For such a graph F, a classical result of Simonovits from 1966 shows that every graph on [ABSTRACT FROM AUTHOR]

Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem. We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices. [ABSTRACT FROM AUTHOR]

Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. In particular, the theorem of Schacht yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdős and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting. We then use this theorem to derive several new results, among them a random version of the Erdős-Simonovits stability theorem for arbitrary graphs, extending the result of Conlon and Gowers, who proved such a statement for so-called strictly 2-balanced graphs. The main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 269-289, 2014 [ABSTRACT FROM AUTHOR]

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turán's theorem, which deals with graphs G = ([ n], E) such that no member of the restriction set \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal {R}\end{align*} \end{document} = \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*} \end{document} induces a copy of K r. Firstly, we examine what happens when this restriction set is replaced by \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal {R}\end{align*} \end{document} = { X∈ \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*} \end{document}: X ∩ [ m]≠ ∅︁}. That is, we determine the maximal number of edges in an n -vertex such that no K r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r -uniform hypergraph \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal R\end{align*} \end{document} on vertex set [ n] is called Turánnical (respectively ε - Turánnical), if for any graph G on [ n] with more edges than the Turán number t r( n) (respectively (1 + ε) t r( n) ), no hyperedge of \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal {R}\end{align*} \end{document} induces a copy of K r in G. We determine the thresholds for random r -uniform hypergraphs to be Turánnical and to be ε -Turánnical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Łuczak-Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 [ABSTRACT FROM AUTHOR]

Abstract: In this note we complete an investigation started by Erdős in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán’s theorem in extremal graph theory. Let be the complete -partite graph with parts of sizes with an edge added to the first part. Letting be the number of edges of the -partite Turán graph of order , we prove that: For all and all sufficiently small , every graph of sufficiently large order with edges contains a . We also give a corresponding stability theorem and two supporting results of wider scope. [Copyright &y& Elsevier]

Abstract: This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following analogue for transitive closure in digraphs of Radon''s classical convexity theorem [J. Radon, Mengen konvexer Körper, die einer gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113–115]: in a complete digraph on vertices with arcs, it is possible to partition the arc set into two subsets whose transitive closures have an arc in common. [Copyright &y& Elsevier]

Abstract: Let A be an complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting be the Frobenius norm of A, we show that This inequality is tight within an additive term . If the matrix A is Hermitian, then This inequality is sharp; moreover, it implies the Turán theorem for graphs. [Copyright &y& Elsevier]

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K 5-minor or no K 3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8( n − 2). [ABSTRACT FROM AUTHOR]

Abstract: The generalized Turán number of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph on m vertices, the value of is , where as , by the Erdős–Stone–Simonovits theorem. In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs ''85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307–331. MR 89d:05158]. A graph G is -bi-jumbled if for every two sets of vertices . Here is the number of pairs such that , , and . This condition guarantees that G and the binomial random graph with edge probability q share a number of properties. Our results imply that, for example, for any triangle-free graph H with maximum degree Δ and for any there exists so that the following holds: any large enough m-vertex, -bi-jumbled graph G satisfies [Copyright &y& Elsevier]

Abstract: Erdős posed the problem of finding conditions on a graph G that imply , where is the largest number of edges in a triangle-free subgraph and is the largest number of edges in a bipartite subgraph. Let be the least number so that any graph G on n vertices with minimum degree has . Extending results of Bondy, Shen, Thomassé and Thomassen we show that . [Copyright &y& Elsevier]

Abstract: Fix . Let be the -uniform hypergraph obtained from the complete graph by enlarging each edge with a set of new vertices. Thus has vertices and edges. We prove that the maximum number of edges in an -vertex -uniform hypergraph containing no copy of isas . This is the first infinite family of irreducible -uniform hypergraphs for each odd whose Turán density is determined. Along the way, we give three proofs of a hypergraph generalization of Turán''s theorem. We also prove a stability theorem for hypergraphs, analogous to the Simonovits stability theorem for complete graphs. [Copyright &y& Elsevier]

Abstract: Brualdi and Mellendorf raised the following problem as a modification of Turán''s theorem. Let t and n be positive integers with . Determine the maximum number of edges of a graph of order n that contains neither nor as a subgraph. They solved it for (see R.A. Brualdi, S. Mellendorf, Two extremal problems in graph theory, The Electron. J. Combin. 1 (1994) #R2). We consider the following similar modification of Turán''s theorem. Let t and n be positive integers with . Determine the maximum number of edges of a graph of order n that contains neither nor as a subgraph. We solve it for . [Copyright &y& Elsevier]

Tura´n's theorem (Mat. Fiz. Lapok 48 (1941) 436) (or rather its extension by Zykov (Mat. Sbornik 24 (66) (1949) 163) answers the following question: For k=2,…,r, what is the maximum number of k-cliques (i.e., subgraphs on k vertices) in a finite graph G, given the clique number r and the number of vertices of G? Here we address—and answer—the following closely related question: For k=3,…,r, what is the maximum number of k-cliques in G, given the clique number r and the number of edges of G? We also prove a “stability theorem” which shows that our result is best possible in a strong sense. [Copyright &y& Elsevier]

Turán's theorem is a cornerstone of the extremal graph theory. In this paper, we determine the minimum number of edges of a connected graph without containing an independent vertex set of a given size and give a new proof of Turán's theorem. [ABSTRACT FROM AUTHOR]