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On a Problem of Graph Theory
... Any graph on n vertices with ex(n, G) edges and which has no copy of G as a subgraph is called extremal. Of particular interest is the behavior of ex(n, C 4 ) where C 4 denotes the cycle of length 4. In [7], Erdős, Rényi and Sós proved that ex(n, C 4 ) ∼ 1 2 n 3/2 using the graphs ER q for constructive lower bounds. Füredi later demonstrated in [8] and [9] that the graphs ER q are extremal when q is even or q > 13. ...
... Therefore, an absolute point is never adjacent to an internal or another absolute point. On the other hand, an internal point of this set is not adjacent to any other internal point, as shown in Theorem 2. 7. Therefore, the set under consideration is indeed an independent set in ER q . ...
We investigate the independence number of two graphs constructed from a polarity of . For the first graph under consideration, the Erd\H{o}s-R\'enyi graph , we provide an improvement on the known lower bounds on its independence number. In the second part of the paper we consider the Erd\H{o}s-R\'enyi hypergraph of triangles . We determine the exact magnitude of the independence number of , q even. This solves a problem posed by Mubayi and Williford.
... The use of finite geometry to construct graphs with interesting properties has a rich history in graph theory. One of the most well-known constructions is due to Brown [6], and Erdős, Rényi, and Sós [11] who used an orthogonal polarity of a Desarguesian projective plane to give examples of graphs that give an asymptotically tight lower bound on the Turán number of the 4-cycle. Later these same graphs were used to solve other extremal problems in a variety of areas such as Ramsey theory [3], [15], hypergraph Turán theory [16], and even the Cops and Robbers game on graphs [5]. ...
... Later these same graphs were used to solve other extremal problems in a variety of areas such as Ramsey theory [3], [15], hypergraph Turán theory [16], and even the Cops and Robbers game on graphs [5]. While our focus is not on the graphs of [6] and [11], we take a moment to define them. Let q be a power of a prime and P G(2, q) be the projective geometry over the 3-dimensional vector space F 3 q . ...
Given a projective plane and a polarity of , the corresponding polarity graph is the graph whose vertices are the points of , and two distinct points and are adjacent if is incident to in . A well-known example of a polarity graph is the Erd\H{o}s-R\'{e}nyi orthogonal polarity graph , which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of , the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order q has an independent set of size . Some related results are also obtained.
... Therefore, we can use Proposition 6.5 to show concentration around the mean in this interval. In particular, we can apply Theorem 5.5 to obtain the 0-statement in the region defined by (11) and (20) as there is an asymptotic gap between the two by (19), and additionally (21) holds. ...
... It is conjectured that this upper bound gives the correct order, but the problem of finding lower bounds (namely, explicit constructions) has been shown to be more difficult. Some instances are known to close the gap in the order of magnitude: Erdős, Rényi and Sós [11] found a lower bound for zex(n, K 2,t ) which match the upper bound, Brown [6] and Füredi [14] proved the right order of magnitude for zex(n, K 3,3 ). Finally the case zex(n, K s,t ) with s ≥ (t − 1)! + 1 was proved by Alon, Rónyai and Szabó [1]. ...
We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.
... To determine z(m, n; 2, 2) is a notorious open (instance of) Zarankievicz's problem. When n is sufficiently large, the value z(n, n; 2, 2) can be bounded as follows [3,17]: ...
... Given Theorem 3.7 on the equivalence of µ t (L(K n )) and ex(n; C 4 ), the upper bound was first proved by Reiman in [32]. The lower bound (and a rediscovery of the upper bound) can be found in [3] and [17]. □ We next proceed with finding similar results as the above ones for the other two remaining mutual-visibility parameters (outer and dual). ...
The mutual-visibility problem in a graph G asks for the cardinality of a largest set of vertices S ⊆ V (G) so that for any two vertices x, y ∈ S there is a shortest x, y-path P so that all internal vertices of P are not in S. This is also said as x, y are visible with respect to S, or S-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside S. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size. *
... The topology considered in this paper is the friendship graph with the set of edges among friends as the fix edges. A friendship graph is a graph where every pair of vertices has exactly one common neighbor [6]. Friendship graphs, due to their properties have been utilized in different disciplines, including block code design [15], set theory [16] and graph labeling [8]. ...
A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot's consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.
... The friendship (or Dutch-Windmill) graph F n is a graph that can be constructed by coalescence n copies of the cycle graph C 3 of length 3 with a common vertex. The Friendship Theorem of Paul Erdös, Alfred Rényi and Vera T. Sós [12], states that graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. Figure 1 shows some examples of friendship graphs. ...
Let be a simple graph of order n. The total dominating set of G is a subset D of V that every vertex of V is adjacent to some vertices of D. The total domination number of G is equal to minimum cardinality of total dominating set in G and denoted by . The total domination polynomial of G is the polynomial , where is the number of total dominating sets of G of size i. In this paper, we study roots of total domination polynomial of some graphs. We show that all roots of lie in the circle with center and the radius , where is the minimum degree of G. As a consequence we prove that if , then every integer root of lies in the set .
... While this arose as a problem in extremal graph theory, the best constructions come from finite geometry and use projective planes and difference sets. Roughly 30 years later, Brown [3], and Erdős, Rényi, and Sós [8,9] independently showed that ex(n, C 4 ) = 1 2 n 3/2 + o(n 3/2 ). They constructed, for each prime power q, a C 4 -free graph with q 2 + q + 1 vertices and 1 2 q(q + 1) 2 edges. ...
Let F be a graph, be an integer, and write for the maximum number of edges in an n-vertex graph that is k-partite and has no subgraph isomorphic to F. The function has been studied by many researchers. Finding is a special case of the Zarankiewicz problem. We prove an analogue of the K\"{o}v\'{a}ri-S\'{o}s-Tur\'{a}n Theorem for 3-partite graphs by showing for . Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case that and is odd, i.e., for . In the cases of and , we use a result of Allen, Keevash, Sudakov, and Verstra\"{e}te, to show that a similar upper bound holds for all , and gives a better constant when s=t=3. Lastly, we point out an interesting connection between difference families from design theory and .
... Nowadays, this result is known as the Friendship theorem. As mentioned in the introduction, the first proof (by contradiction) was given by Erdős, Rényi and Sós [12] in 1966, and is considered to be the most successful. Basically, it has two parts: First, it is proved that if the graph G which models such a cocktail party (where people correspond to vertices and friendships are represented by edges) is a counterexample with more than three vertices, then it has to be regular, say with degree k. ...
As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to relationships between people or groups of people. In this article, we comment on four results of this kind, which are related to various general theories on graphs and their applications: the Handshake lemma (related to graph colorings and Boolean algebra), a lemma on known and unknown people at a cocktail party (to Ramsey theory), a theorem on friends in common (to distance-regularity and coding theory), and Hall's Marriage theorem (to the theory of networks). These four areas of graph theory, often with problems which are easy to state but difficult to solve, are extensively developed and currently give rise to much research work. As examples of representative problems and results of these areas, which are discussed in this paper, we may cite the following: the Four Colors Theorem (4CTC), the Ramsey numbers, problems of the existence of distance-regular graphs and completely regular codes, and finally the study of topological proprieties of interconnection networks.
... If H (n, ℓ, d) is empty, then, following the usual convention, we shall write e(n, ℓ, d) = ∞. For more details on this problem, we refer to [3,4,18,19]. ...
The Steiner diameter of a graph G, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When k=2, is the classical diameter. The problem of determining the minimum size of a graph of order n whose diameter is at most d and whose maximum is was first introduced by Erd\"{o}s and R\'{e}nyi. Recently, Mao considered the problem of determining the minimum size of a graph of order n whose Steiner k-diameter is at most d and whose maximum is at most , where , and studied this new problem when k=3. In this paper, we investigate the problem when .
... This paper shows examples of such topologies based on incidence graphs of projective planes and compares them with competitive alternatives. Incidence graphs of finite projective planes [7], [16] have been used to attain the Moore bound, but not only mathematicians have paid attention to this discrete structures. In fact, Valerio et al. already use them to define Orthogonal Fat Trees (OFT) [39], which are highly scalable cost optimal indirect networks. ...
The interconnection network comprises a significant portion of the cost of large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with the aim of minimizing these costs. However, they fail to consider potential unbalance in the network utilization, which in some cases results in suboptimal designs. Based on an appropriate cost model, this paper advocates the use of networks based on incidence graphs of projective planes, broadly denoted as Projective Networks. Projective Networks rely on highly symmetric generalized Moore graphs and encompass several proposed direct (PN and demi-PN) and indirect (OFT) topologies under a common mathematical framework. Compared to other proposals with average distance between 2 and 3 hops, these networks provide very high scalability while preserving a balanced network utilization, resulting in low network costs. Overall, Projective Networks constitute a competitive alternative for exascale-level interconnection network design.
... In this case, a famous result of Kővári, Sós and Turán [15] shows that ex(n, K s,t ) = O s,t (n 2−1/s ) whenever s ≤ t. This bound was shown to be tight for s = 2 by Esther Klein [6] (see also [3,8]) and for s = 3 by Brown [3]. For higher values of s, it is only known that the bound is tight when t is sufficiently large in terms of s. ...
Given a family of graphs , the extremal number is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs such that . This solves a longstanding problem in the area of extremal graph theory.
... Thus our graph has q + 1 loops. The Erdős-Rényi graphs are of interest because they do not contain any 4-cycles, but nonetheless they have a large number of edges; this is the motivation for [4]. For further work on these graphs, see [8,11]. ...
We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd\H{o}s-R\'{e}nyi graphs. We investigate further properties of our bounds, and show how our results on the Erd\H{o}s-R\'{e}nyi graphs can be extended to other polarity graphs.
... The famous Friendship Theorem of Erdős, Rényi, and Sós ( [6], Theorem 6) states that if, in a finite undirected graph G, every two vertices have precisely one common neighbour, then some vertex of G is adjacent to all the vertices of G except itself. ...
We show that the method of counting closed walks in strongly regular graphs rules out no parameter sets other than those ruled out by the method of counting eigenvalue multiplicities.
... Nevertheless, the upper bound is tight for t = 2, as shown by Erdős, Rényi and Sós ( [7]), and t = 3, as shown by Brown ([4]). On the other hand, the asymptotic behaviour of ex(n, K t,t ) is not known for any t ≥ 4. A significant step towards a solution to this problem was made by Kollár, Rónyai and Szabó ( [10]), who constructed the norm graphs G(q, t), a family of graphs which are extremal for K t,s , for any s ≥ t! + 1. ...
The projective norm graphs P(q, 4) introduced by Alon, R\'onyai and Szab\'o are explicit examples of extremal graphs not containing K_4,7. Ball and Pepe showed that P(q, 4) does not contain a copy of K_5,5 either for q >= 7, asymptotically improving the best lower bound for ex(n, K_5,5). We show that these results can not be improved, in the sense that P(q, 4) contains a copy of K_4,6 for infinitely many primes q.
... Another natural measure is the diameter of G, that is, the maximum distance between any pair of vertices. The goal then becomes to reduce the diameter as much as possible by adding a given number of edges, or to achieve a given diameter with a small number of extra edges; see for example the papers by Erdös, Rényi, and Sós [6,7]. ...
Let C be the unit circle in . We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place \emph{shortcuts} on C such that the diameter of the resulting graph is minimized. We analyze for each k with what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of~k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is for any~k.
... When k = 3, we have b 1 +b 2 +b 3 ≤ n, so it is clear that N * ≤ 3 i=1 ex(b i , H) ≤ ex(n, H). When H = C 4 , using the well-known result (see [6,2,11]) that ex(n, C 4 ) = (1/2 + o(1)) · n 3/2 , it holds that ...
Let H be a fixed graph. Denote f(n,H) to be the maximum number of edges not contained in any monochromatic copy of H in a 2-edge-coloring of the complete graph , and ex(n,H) to be the {\it Tur\'an number} of H. An easy lower bound shows for any H and n. In \cite{KS2}, Keevash and Sudakov proved that if H is an edge-color-critical graph or , then holds for large n, and they asked if this equality holds for any graph H when n is sufficiently large. In this paper, we provide an affirmative answer to this problem for an abundant infinite family of bipartite graphs H, including all even cycles and complete bipartite graphs for or . In addition, our proof shows that for all such H, the 2-edge-coloring c of achieves the maximum number f(n,H) if and only if one of the color classes in c induces an extremal graph for ex(n,H). We also obtain a multi-coloring generalization for bipartite graphs. Some related problems are discussed in the final section.
... If H (n, ℓ, d) is empty, then, following the usual convention, we shall write e(n, ℓ, d) = ∞. For more details on this problem, we refer to [5,6,18,19]. ...
The Steiner k-diameter of a graph G, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When k=2, is the classical diameter. The problem of determining the minimum size of a graph of order n whose diameter is at most d and whose maximum is was first introduced by Erd\"{o}s and R\'{e}nyi. In this paper, we generalize the above problem for Steiner k-diameter, and study the problem of determining the minimum size of a graph of order n whose Steiner 3-diameter is at most d and whose maximum is at most .
... For k = 2, this reduces to a well-known problem of studying the Turán number for the 4-cycle. It is known that f 2 (n) = (1+o(1))n 3/2 [2,4] and the exact value of f 2 (n) for infinitely many n is obtained in [6]. For k ≥ 3, Füredi [7] showed that n−1 k−1 + ⌊ n−1 k ⌋ ≤ f k (n) ≤ 7 2 n k−1 and conjectured the following. ...
Let be an odd integer and let n be a sufficiently large integer. We prove that the maximum number of edges in an n-vertex k-uniform hypergraph containing no 2-regular subgraphs is , and the equality holds if and only if H is a full k-star with center v together with a maximal matching omitting v. This verifies a conjecture of Mubayi and Verstra\"{e}te.
... Extremal problems on the diameter of graphs have a long history and were first investigated by Erdős and Rényi [5] and Erdős, Rényi, and Sós [6], who studied the minimum number of edges in an n-vertex graph with diameter at most d. Another line of research concerning the change of diameter if edges of the graph are added or deleted was initiated by Chung and Garey [2] (see also, e.g., [3,1]). ...
We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E) on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that |E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma, Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a constant depending only \epsilon. This shows that every dense graph can be partitioned into a small number of ``small worlds'' provided that few edges can be ignored. Improving on their result, we determine P(n,\epsilon,d) within an absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded for \epsilon < 1/4, P(n,\epsilon,3) = \Theta(1/\epsilon^2) for \epsilon > n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, R\"odl, Ruci\'nski, and Szemer\'edi.
... The upper bound of [8] follows directly from the observation that a natural greedy construction never closes a cycle with length at most α + 1; the lower bound follows from the fact that no strict subgraph of a graph with girth α + 2 is an α-spanner. 1 It has been conjectured [32,17,15] that the trivial upper bound m 2k+1 (n), m 2k+2 (n) = O(n 1+1/k ) is sharp up to the leading constant, but this Girth Conjecture has only been proved for k = 1 (trivial), and k ∈ {2, 3, 5} [18,33,49,60,58,12,39]. See [40,41,61] for lower bounds on m g (n). ...
Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say bits. There is an inherent tradeoff between the sparsity parameter and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. In this paper we show that the recent additive spanner lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that fully characterize the asymptotic behavior of the optimal stretch function f as a function of . Specifically, for any integer , any compression scheme with size has a sublinear additive stretch function f: This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's -spanners have an essentially optimal tradeoff between and , and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of -spanners with size , where . This size bound improves on the spanners of Elkin and Peleg (2004), Thorup and Zwick (2006), and Pettie (2009). According to our lower bounds neither the size nor stretch function can be substantially improved.
... A well-known theorem of Kővári, Sós and Turán [19] shows that ex(n, K s,t ) = O(n 2−1/s ) for any integers t ≥ s. For s = 2, 3, matched lower bounds were found in [11,5] respectively; for other values of s, this bound was known to be tight when t is sufficiently larger than s, which was first proved by Kollár, Rónyai and Szabó [18] and then slightly improved to t > (s − 1)! by Alon, Rónyai and Szabó [1]. Recently, Blagojević, Bukh and Karasev [3] and Bukh [6] used the random algebraic method to give different constructions which yield the same lower bound ex(n, K s,t ) = Ω(n 2−1/s ) as in [18,1], provided that t is sufficiently large. ...
Let be the complete r-partite r-uniform hypergraph and be the maximum number of edges in any n-vertex -free r-uniform hypergraph. It is well-known in the graph case that when t is sufficiently larger than s. In this note, we generalize the above to hypergraphs by showing that if is sufficiently larger than then This follows from a more general Tur\'an type result we establish in hypergraphs, which also improves and generalizes some recent results of Alon and Shikhelman. The lower bounds of our results are obtained by the powerful random algebraic method of Bukh. Another new, perhaps unsurprising insight which we provide here is that one can also use the random algebraic method to construct non-degenerate (hyper-)graphs for various Tur\'an type problems. The asymptotics for is also proved by Verstra\"ete independently with a different approach.
... For example, the Even Cycle Problem proposed by Erdős [Erd64a,BS74], which asks for the exponent of ex(n, C 2k ), is still open for every k not in {2, 3, 5} (see e.g. [ERS66,Ben66,Wen91,LU93,LUW99]). For more results on degenerate Turán problems, we refer the reader to the survey [FS13]. ...
For a positive real number p, the p-norm of a graph G is the sum of the p-th powers of all vertex degrees. We study the maximum p-norm of F-free graphs on n vertices, focusing on the case where F is a bipartite graph. The case corresponds to the classical degenerate Tur\'{a}n problem, which has yielded numerous results indicating that extremal constructions tend to exhibit certain pseudorandom properties. In contrast, results such as those by Caro--Yuster, Nikiforov, and Gerbner suggest that for large p, extremal constructions often display a star-like structure. It is natural to conjecture that for every bipartite graph F, there exists a threshold such that for , the order of is governed by pseudorandom constructions, while for , it is governed by star-like constructions. We confirm this conjecture by determining the exact value of , under a mild assumption on the growth rate of . Our results extend to r-uniform hypergraphs as well. We also prove a general upper bound that is tight up to a factor for when . We conjecture that this factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles. Our proofs involve p-norm adaptions of fundamental tools from degenerate Tur\'{a}n problems, including the Erd\H{o}s--Simonovits Regularization Theorem and the Dependent Random Choice.
... Let C 2ℓ be the cycle of length 2ℓ. While the classical upper bound expn, C 2ℓ q " O ℓ pn 1`1{ℓ q has been proved by Bondy and Simonovits [5] fifty years ago, the matching lower bounds are known to exist only for C 4 , C 6 and C 10 , see [7,10,20]. Whether expn, C 2ℓ q " Ω ℓ pn 1`1{ℓ q holds for ℓ R t2, 3, 5u has remained a long-standing open problem. For complete bipartite graphs K s,t with s ď t, the well-known Kővári-Sós-Turán theorem [17] states that expn, K s,t q " O s,t pn 2´1{s q. ...
A graph G is (c,t)-sparse if for every pair of vertex subsets with , . In this paper we prove that for every and integer , there exists such that if an n-vertex graph G is (c,t)-sparse for some t, and has at least edges, then G contains an induced copy of . This resolves a conjecture of Fox, Nenadov and Pham.
... A graph G ∈ G n is said to be (2, 3)-orientable if some orientation of the edges of G yeilds a digraph that is (2, 3)-cordial. Some interest has lately been shown in the cordiality of the friendship graphs, introduced by Erdős, Rényi and Sós, [3], with a proof of the friendship theorem: If any two distinct vertices of a simple graph have exactly one common neighbor then some vertex is adjacent to all other vertices. Such graphs are known as friendship graphs, F n on 2n + 1 vertices. ...
A (0; 1)-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let g be a labeling of the edge set of a graph that is induced by a labeling f of the vertex set. If both g and f are friendly then f is said to be a cordial labeling of the graph. This concept extended to directed graphs is called (2; 3)-cordiality of digraphs. We investigate the labelings that are both cordial for a graph and (2; 3)-cordial for an orientation of it. We also consider the same problem for other known binary vertex labelings of graphs.
... Let F n = F n, n−1 2 . For odd n, F n is known as the friendship graph (or windmill graph) due to the Friendship Theorem [6]: a graph in which any two distinct vertices have exactly one common neighbor has a vertex joined to all others. The main results are as follows. ...
Answers are offered to Gould’s question to find spectral sufficient conditions that imply a graph contains a chorded cycle via signless Laplacian spectral radius. The conditions are tight.
... Brown [4], and Erdős, Rényi and Sós [14] independently and simultaneously proved that ex(q 2 + q + 1, C 4 ) ≥ 1 2 q(q + 1) 2 for all prime powers q. Füredi [8,9] proved that for all prime powers q ≥ 14, ex(q 2 + q + 1, C 4 ) = 1 2 q(q + 1) 2 . ...
Let be a family of graphs. A graph G is -free if G does not contain any as a subgraph. The Tur\'an number is the maximum number of edges in an n-vertex -free graph. Let be the matching consisting of s independent edges. Recently, Alon and Frank determined the exact value of . Gerbner obtained several results about when F satisfies certain proportions. In this paper, we determine the exact value of when are large enough for every . When n is large enough, we also show that for and when and s is large enough.
... (To determine the value z(m, n; 2, 2) is an instance of the Zarankiewicz's problem, see [25].) When n is sufficiently large, the value z(n, n; 2, 2) can be bounded as follows [4,13]: ...
Given a graph G, a mutual-visibility coloring of G is introduced as follows. We color two vertices with a same color, if there is a shortest x,y-path whose internal vertices have different colors than x,y. The smallest number of colors needed in a mutual-visibility coloring of G is the mutual-visibility chromatic number of G, which is denoted . Relationships between and its two parent ones, the chromatic number and the mutual-visibility number, are presented. Graphs of diameter two are considered, and in particular the asymptotic growth of the mutual-visibility number of the Cartesian product of complete graphs is determined. A greedy algorithm that finds a mutual-visibility coloring is designed and several possible scenarios on its efficiency are discussed. Several bounds are given in terms of other graph parameters such as the diameter, the order, the maximum degree, the degree of regularity of regular graphs, and/or the mutual-visibility number. For the corona products it is proved that the value of its mutual-visibility chromatic number depends on that of the first factor of the product. Graphs G for which are also considered.
... It is a classical result by Erdős that such tournaments exist [5]. By applying a probabilistic method, he obtained that there areparadoxical tournaments of size O ( 2 2 ); also, he provided a lower bound Ω(2 ). ...
Maslov's class is an expressive fragment of First-Order Logic known to have decidable satisfiability problem, whose exact complexity, however, has not been established so far. We show that has the exponential-sized model property, and hence its satisfiability problem is NExpTime-complete. Additionally, we get new complexity results on related fragments studied in the literature, and propose a new decidable extension of the uniform one-dimensional fragment (without equality). Our approach involves a use of satisfiability games tailored to and a novel application of paradoxical tournament graphs.
... Neither graphs nor digraphs in this paper have loops, multiple edges, or multiple arcs. In 1966, Erdös et al. [9] introduced and proved the Friendship Theorem. The Friendship Theorem can be stated, in graph-theoretical terms, as follows: if any pair of vertices in a graph has exactly one common neighbor, then there exists a vertex adjacent to the others. ...
In this paper, we introduce the notion of two-way -liking digraphs as a way to extend the results for generalized friendship graphs. A two-way -liking digraph is a digraph in which every t vertices have exactly common out-neighbors and common in-neighbors. We first show that if , then a two-way -liking digraph of order n is k-diregular for a positive integer k satisfying the equation . This result is comparable to the result by Bose and Shrikhande in 1969 and actually extends it. Another main result is that if , then the complete digraph on vertices is the only two-way -liking digraph. This result can stand up to the result by Carstens and Kruse in 1977 and essentially extends it. In addition, we find that two-way -liking digraphs are closely linked to symmetric block designs and extend some existing results of -liking digraphs.
... (1) It is known that ex(n; C 3 , C 4 ) = (0.5 + o(1))n 3/2 as n → ∞ (see [8]). Similarly to the proof of Lemma 18, it can be proved that there exists a {C 3 , C 4 }free bipartite graph G = (V, E), where |V | = n and |E| ≫ n 3/2 . ...
Let be the maximal size of a set such that the equation has no solution with and integer x. Erd\H{o}s, S\'ark\"ozy and T. S\'os studied , and gave bounds when k=2,3,4,6 and also in the general case. We study the problem for d=3, and provide bounds for k=2,3,4,6 and 9, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstra\"ete. We also introduce another function closely related to : While the original problem requires to all be distinct, we can relax this and only require that the multiset of the 's cannot be partitioned into d-tuples where each d-tuple consists of d copies of the same number.
... For the upper bound of (3), we use the well-known result [17] ex(n, C 4 ) = ( 1 2 + o(1))n 3/2 . Applying Proposition 15, we obtain that a constant c 2 > 1 √ 2 can be chosen if r is large enough. ...
In this paper, we address problems related to parameters concerning edge mappings of graphs. Inspired by Ramsey's Theorem, the quantity m(G,H) is defined to be the minimum number n such that for every f:E(Kn)→E(Kn) either there is a fixed copy of G with f(e)=e for all e∈E(G), or a free copy of H with f(e)∉E(H) for all e∈E(H). We extend many old results from the 80's as well as proving many new results. We also consider several new interesting parameters with the same spirit.
... The Erdős-Rényi graphs, have vertices as the points of PG(2, q), and u is adjacent to v if u T v = 0, where we identify vertices with 1-dimensional subspaces of GF(q) 3 . These are well-known examples of graphs which are C 4 -free extremal, in the sense that they have the largest possible number of edges in a C 4 -free graph on q 2 + q + 1 vertices; see [15,19]. For more on polarity graphs, see [28]. ...
We consider the localization number and metric dimension of certain graphs of diameter 2, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of 2, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter 2 and polarity graphs.
... Erdős-Rényi-Sós [7] and Brown [4] showed that ex(q 2 + q + 1, C 4 ) ≥ 1 2 q(q + 1) 2 for all prime powers q. Füredi [13,14] proved that ex(q 2 +q +1, C 4 ) = 1 2 q(q +1) 2 for all prime powers q ≥ 14. ...
Let be a partition with and define . It is easy to check that the trace satisfies for all 4-sets . In the present paper, we prove that if satisfies and , then for some , . Several further results of a similar flavor are established as well.
... For sharp lower bounds, see Erdős, Rényi and Sós [9] for s = t = 2, Axenovich, Füredi and Mubayi [2] for s ≥ t = 2, Brown [8] for s = t = 3, Kollár, Rónyai and Szabó [15], and Alon, Rónyai and Szabó [1] for s = t = 3 and s ≥ (t − 1)! + 1. For t ≥ 4, Bohman and Keevash [4] obtained a lower bound for ex(n, K t,t ) as Ω(n 2−2/(t+1) (log n) 1/(t 2 −1) ). ...
For graphs , the k-color Turán number is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic in color i as a subgraph, where . In this note, we show that if is a bipartite graph with at least two edges for , then as , in which the non-constructive proof for some cases can be derandomized.
... Beginning in Section 6, we will examine Z m -cordiality for the friendship graphs. It is a theorem of Erdős, Rényi, and Sós [8] that finite graphs in which every two vertices possess exactly one neighbor in common are exactly the friendship graphs. ...
... A fundamental result of Bondy and Simonovits [3] shows that ex(n, C 2ℓ ) = O(n 1+1/ℓ ), where C 2ℓ denotes a cycle of length 2ℓ. Matching lower bound constructions are only known for cycles of length four, six and ten [5,17,41]. In particular, it is known that ex(n, C 4 ) = Θ(n 3/2 ). ...
In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an n-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound holds since the complete bipartite graph can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant C such that every n-vertex with edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without a four-cycle based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an `almost-spanning' robust expander which might be of independent interest.
... The problem of octahedron, dodecahedron and icosahedron graphs were later resolved by Erdős and Simonovits [11] and by Simonovits [19,20] respectively; while the innocent looking cube graph remains elusive. Two basic classes of bipartite graphs with high symmetry are even cycles and complete bipartite graphs; both of them have been widely studied for several decades [2,3,5,6,8,15,17,22]. For more on the bipartite Turán problem, we refer the reader to the comprehensive survey of Füredi and Simonovits [13]. ...
We study the Tur\'{a}n problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. \begin{enumerate} \item The prism is the graph consisting of two vertex disjoint -cycles and a matching pairing the corresponding vertices of these two cycles. We show that for every , ex. This resolves a conjecture of He, Li and Feng. \item The hexagonal tiling in honeycomb is one of the most natural structures in the real world. We show that the extremal number of honeycomb graphs has the same order of magnitude as their basic building unit 6-cycles. \item We also consider bipartite graphs from quadrangulations of the cylinder and the torus. We prove near optimal bounds for both configurations. In particular, our method gives a very short proof of a tight upper bound for the extremal number of the 2-dimensional grid, improving a recent result of Brada\v{c}, Janzer, Sudakov and Tomon. \end{enumerate} Our proofs mix several ideas, including shifting embedding schemes, weighted homomorphism and subgraph counts and asymmetric dependent random choice.
The multicolor Ramsey number rk(C4) is the smallest integer N such that any k-edge coloring of KN contains a monochromatic C4. The current best upper bound of rk(C4) was obtained by Chung (1974) and independently by Irving (1974), i.e., rk(C4) ≤ k2 + k + 1 for all k ≥ 2. There is no progress on the upper bound since then. In this paper, we improve the upper bound of rk(C4) by showing that rk(C4) ≤ k2 + k − 1 for even k ≥ 6. The improvement is based on the upper bound of the Turán number ex(n, C4), in which we mainly use the double counting method and many novel ideas from Firke, Kosek, Nash, and Williford [J. Combin. Theory, Ser. B 103 (2013), 327–336].
Sufficient conditions for the existence of various kinds of subgraphs in a graph have been extensively studied. Recently, Gould (2022) studied the existence of a chorded cycle in a graph form spectral perspectives and proposed a question that "What spectral conditions imply the existence of a chorded cycle in a graph?" Zheng et al. (2023) and Xu et al. (2023) respectively gave the answers to this question by providing the sufficient conditions involving the spectral radius and the signless Laplacian spectral radius for the existence of a chorded cycle in a graph. In this paper, we generalize Xu et al.'s result to λ α 1 (G) for α ∈ [ 1 2 , 1), where λ α 1 (G) is the spectral radius of A α (G) := α D(G) + (1 − α)A(G), D(G) and A(G) being respectively the adjacency matrix and the diagonal degree matrix of G.
In 1962, Erd\H{o}s proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem in 1963. In this paper we establish several spectral analogues of Moon and Moser's theorem on Hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs. One main ingredient of our proofs is a structural result of its own interest, involving Hamilton paths in balanced bipartite graphs with given minimum degree and number of edges.
In the Constructor–Blocker game, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph . For given graphs and , Constructor can only claim edges that leave her graph ‐free, while Blocker has no restrictions. Constructor's goal is to build as many copies of as she can, while Blocker attempts to minimize the number of copies of in Constructor's graph. The game ends once there are no more edges that Constructor can claim. The score of the game is the number of copies of in Constructor's graph at the end of the game when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on to many pairs of and : We determine when and , also when both and are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of when and .
The Turán problem asks for the largest number of edges ex(n, H) in an n-vertex graph not containing a fixed forbidden subgraph H, which is one of the most important problems in extremal graph theory. However, the order of magnitude of ex(n, H) for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomial resultant approach to the algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lower bounds on the Turán number of 1-subdivision of and the linear Turán number of the Berge theta hypergraph , where t1 = 25 and t2 = 217. Moreover, the constant t1 improves the random algebraic construction of Bukh and Conlon (2018) and makes the known estimation better on the smallest value of t1 concerning a problem posed by Conlon et al. (2021) by reducing the value from a magnitude of 1056 to the number 25, while the constant t2 improves a result of He and Tait (2019).
We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an r-graph F is bounded if there exist constants such that for large n, every n-vertex F-free r-graph with a vertex of degree at least has fewer than edges. The boundedness property is crucial for recent works~\cite{HHLLYZ23a,DHLY24} that aim to extend the classical Hajnal--Szemer\'{e}di Theorem and the anti-Ramsey theorems of Erd\H{o}s--Simonovits--S\'{o}s. We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for 3-graphs, strengthening a theorem by Kostochka--Mubayi--Verstra\"{e}te~\cite{KMV15}.
In this paper we study a multi-partite version of the Erd\H{o}s--Stone theorem. Given integers and , let be the maximum number of edges of -free k-partite graphs with n vertices in each part, where is the t-blowup of . An easy consequence of the supersaturaion result gives that . Similar to a result of Erd\H os and Simonovits for the non-partite case, we find that the error term is closely related to the (multi-partite) Zarankiewicz problem. Using such Zarankiewicz numbers, for t=2,3, we determine the error term up to an additive linear term; using some natural assumptions on such Zarankiewicz numbers, we determine the error term up to an additive constant depending on k, r and t. We actually obtain exact results in many cases, for example, when . Our proof uses the stability method and starts by proving a stability result for -free multi-partite graphs.
We show that the maximum number of edges in a 3-uniform hypergraph without a Berge cycle of length four is at most . This improves earlier estimates by Győri and Lemons and by Füredi and Özkahya.
We study the hardness of the problem of finding the distance of quantum error-correcting codes. The analogous problem for classical codes is known to be NP-hard, even in approximate form. For quantum codes, various problems related to decoding are known to be NP-hard, but the hardness of the distance problem has not been studied before. In this work, we show that finding the minimum distance of stabilizer quantum codes exactly or approximately is NP-hard. This result is obtained by reducing the classical minimum distance problem to the quantum problem, using the CWS framework for quantum codes, which constructs a quantum code using a classical code and a graph. A main technical tool used for our result is a lower bound on the so-called graph state distance of 4-cycle free graphs. In particular, we show that for a 4-cycle free graph
G
, its graph state distance is either δ or δ + 1, where δ is the minimum vertex degree of
G
. Due to a well-known reduction from stabilizer codes to CSS codes, our results also imply that finding the minimum distance of CSS codes is also NP-hard.
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