Clique Of Order Research Articles

Finding and counting small tournaments in large tournaments

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, we prove that every tournament on four vertices (there are four) can be detected in O(n2) time and counted in O(nω) time where ω<2.372 is the matrix multiplication exponent. We further prove that any tournament on five vertices (there are 12) can be counted in O(nω+1) time. As for lower-bounds, we prove that for almost all k-vertex tournaments, the complexity of the detection problem is not easier than the complexity of the corresponding well-studied counting problem for undirected cliques of order k−O(log⁡k).

Spectral extremal graphs for edge blow-up of star forests

The edge blow-up of a graph G, denoted by Gp+1, is obtained by replacing each edge of G with a clique of order p+1, where the new vertices of the cliques are all distinct. Yuan (2022) [20] determined the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In this paper, we prove that the graphs with the maximum spectral radius in an n-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when n is sufficiently large.

Exact solutions to the Erdős-Rothschild problem

Abstract Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty $ : (i) A sufficient condition on $\boldsymbol {k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case $\boldsymbol {k}=(3,\ldots ,3)$ of length $7$ , the unique extremal graph is the complete balanced $8$ -partite graph, with colourings coming from Hadamard matrices of order $8$ . (iii) In the case $\boldsymbol {k}=(k+1,k)$ , for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$ , the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible.

Solution to a general version of a degree sequence variant of the Erdős–Sós conjecture

Let P denote a graph property, such as “being k-connected”, “being hamiltonian”, or “containing H as a subgraph”, etc. A graphic sequence π is potentially P-graphic if there is a realization of π having the property P. In 1991, Erdős et al. introduced the following problem: Determine the minimum even integer σ(P,n) such that every n-term graphic sequence with sum at least σ(P,n) is potentially P-graphic. The parameter σ(P,n) is known as the potential function of P, and can be viewed as a degree sequence variant of the classical extremal function ex(P,n). A simple graph G is an ℓ-tree if G=Kℓ+1, or G has a vertex whose neighborhood is a clique of order ℓ, and G−v is an ℓ-tree. Clearly, 1-trees are the usual trees. For ℓ≥1 and k≥ℓ+1, let P(ℓ,k) be the graph property “containing every ℓ-tree on k vertices as a subgraph”. For ℓ=1, Yin and Li determined σ(P(1,k),n) for k≥2 and n≥92k2+192k. This is a degree sequence variant of the Erdős–Sós conjecture. For ℓ=2, Zeng et al. determined σ(P(2,k),n) for k≥3 and n sufficiently large. In this paper, we consider the most general case ℓ≥3, and determine σ(P(ℓ,k),n) for ℓ≥3, k≥ℓ+1 and n sufficiently large.

Spectra of quasi-strongly regular graphs

A quasi-strongly regular graph G of grade 2 with parameters (n,k,a;c1,c2) is a k-regular graph on n vertices such that every pair of adjacent vertices have a common neighbors, every pair of distinct nonadjacent vertices have c1 or c2 common neighbors, and for each ci(i=1,2), there exists a pair of non-adjacent vertices sharing ci common neighbors. The children G1 and G2 of the graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G1 or G2 if and only if they share c1 or c2 neighbors, respectively. The graph G is a quasi-strongly regular graph of type A or B if it is of grade 2 and has a child that is a connected strongly regular graph or a disjoint union of cliques of order m(m>1), respectively. In this paper, we investigate the spectral properties of the graph G if it is a quasi-strongly regular graph of type A or B. Several examples of distance-regular graphs which are quasi-strongly regular graphs of type A are given. Moreover, we give several constructions of quasi-strongly regular graphs and calculate their spectrum.

On deeply critical oriented cliques

AbstractIn this work we consider arc criticality in colourings of oriented graphs. We study deeply critical oriented graphs, those graphs for which the removal of any arc results in a decrease of the oriented chromatic number by 2. We prove the existence of deeply critical oriented cliques of every odd order , closing an open question posed by Borodin et al. Additionally, we prove the nonexistence of deeply critical oriented cliques among the family of circulant oriented cliques of even order.

Stability for the Erdős-Rothschild problem

Abstract Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s)$ of natural numbers and a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ , such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of $\log _2 F(n;\boldsymbol {k})/{n\choose 2}$ as n tends to infinity and proved a stability theorem for complete multipartite graphs G. In this paper, we provide a sufficient condition on $\boldsymbol {k}$ which guarantees a general stability theorem for any graph G, describing the asymptotic structure of G on n vertices with $F(G;\boldsymbol {k}) = F(n;\boldsymbol {k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$ . The proof uses a version of symmetrisation on edge-coloured weighted multigraphs.

Extremal graphs for edge blow-up of graphs

Given a graph H and an integer p, the edge blow-up Hp+1 of H is the graph obtained from replacing each edge in H by a clique of order p+1 where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. In this paper, we determine the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In addition, we characterize the extremal graphs for edge blow-up of all non-bipartite graphs. Our results also extend several known results, including the Turán numbers for edge blow-up of stars, paths and cycles. The method we used can also be applied to find a family of counter-examples to a conjecture posed by Keevash and Sudakov in 2004 concerning the maximum number of edges not contained in any monochromatic copy of H in a 2-edge-coloring of Kn.

Clique minors in graphs with a forbidden subgraph

AbstractThe classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at least r has the clique of order r as a minor. Hadwiger's conjecture is an example of a well‐studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on n vertices of independence number at most r. If true Hadwiger's conjecture would imply the existence of a clique minor of order . Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that G is H‐free for some bipartite graph H then one can find a polynomially larger clique minor. This has recently been extended to triangle‐free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph H, answering a question of Dvořák and Yepremyan. In particular, we show that any ‐free graph has a clique minor of order , for some constant depending only on s. The exponent in this result is tight up to a constant factor in front of the term.

On Ramsey Numbers $$R(K_4-e, K_t)$$

Let G and H be finite undirected graphs. The Ramsey number R(G, H) is the smallest integer n such that for every graph F of order n, either F contains a subgraph isomorphic to G or its complement $${\overline{F}}$$ contains a subgraph isomorphic to H. An (s, t)-graph is a graph that contains neither a clique of order s nor an independent set of order t. In this paper we obtain some inequalities involving Ramsey numbers of the form $$R(K_4-e,K_t)$$ . In particular, a constructive proof implies that if G is a $$(k,s+1)$$ -graph, H is a $$(k,t+1)$$ -graph, and both G and H contain a $$(K_k-e)$$ -free graph M as an induced subgraph, then we have $$R(K_{k+1}-e,K_ {s+t+1}) > |V(G)| + |V(H)| + |V(M)|.$$ Furthermore, if $$s \le t$$ , then $$R(K_4-e,K_ {s+t+1}) \ge R(3,s+1)+R(3,t+1)+s$$ . In the experimental part, we use the $$(K_4-e)$$ -free graph generation process to construct graphs witnessing lower bounds for $$R(K_4-e,K_t)$$ , and compare the results obtained by this approach to the results obtained by analogous triangle-free process. Finally, some open problems involving Ramsey numbers of the form $$R(K_4-e,K_t)$$ and their asymptotics are posed.

Turán density of cliques of order five in 3-uniform hypergraphs with quasirandom links

We show that 3-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than 1/3 contain a clique on five vertices. This result is asymptotically best possible.

Many disjoint triangles in co-triangle-free graphs

AbstractWe prove that any n-vertex graph whose complement is triangle-free contains n2/12 – o(n2) edge-disjoint triangles. This is tight for the disjoint union of two cliques of order n/2. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős.

On the subgraph query problem

AbstractGiven a fixed graph H, a real number p (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ &lt; 2, there exists α &lt; 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in nδ queries.

Enumerating cliques in direct product graphs

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs.

Turán’s Theorem for the Fano Plane

Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a complete solution to Tur\'an's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for $n=7$ there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order $7$ by removing all five edges containing a fixed pair of vertices. For sufficiently large values $n$ this was proved earlier by F\"uredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.

The Cyclic Triangle-Free Process

For positive integers s and t, the Ramsey number R ( s , t ) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R ( 3 , t ) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R ( 3 , t ) for small t’s are computed, and R ( 3 , 35 ) ≥ 237 , R ( 3 , 36 ) ≥ 244 , R ( 3 , 37 ) ≥ 255 , R ( 3 , 38 ) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R ( 3 , t ) are proposed.

Max-Bisections of [formula omitted]-free graphs

A bisection of a graph G is a partition of its vertex set into two sets which differ in size by at most 1, and its size is the number of edges between the two sets. The Max-Bisection problem is to find a bisection of G maximizing its size. An (l,r)-fan is a graph on (r−1)l+1 vertices consisting of l cliques of order r that intersect in exactly one common vertex. Let G be a connected graph with n vertices, m edges, and minimum degree at least 2. In this paper, we show that if G contains neither K2,l nor (l,4)-fan, then G has a bisection of size at least m∕2+(n−l+1)∕4, which improves the result given by Jin and Xu. As a corollary, it implies that every connected graph with n vertices, m edges, and minimum degree at least 2 and without cycles of length 4 has a bisection of size at least m∕2+(n−1)∕4. This improves a result given by Fan, Hou and Yu.

Cycle-Complete Ramsey Numbers

Abstract The Ramsey number $r(C_{\ell },K_n)$ is the smallest natural number $N$ such that every red/blue edge colouring of a clique of order $N$ contains a red cycle of length $\ell $ or a blue clique of order $n$. In 1978, Erd̋s, Faudree, Rousseau, and Schelp conjectured that $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell ,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ provided $\ell \geq C\frac{\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon&amp;gt;0$ and $n&amp;gt; n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac{\log n}{\log \log n}$. This proves the conjecture of Erd̋s, Faudree, Rousseau, and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd̋s, Faudree, Rousseau, and Schelp.

Perfect folding of graphs

In this paper we introduced the definition of perfect foldingof graphs and we proved that cycle graphs of even number ofedges can be perfectly folded while that of odd number ofedges can be perfectly folded to C3 . Also we proved thatwheel graphs of odd number of vertices can be perfectlyfolded to C3. Finally we proved that if G is a graph of n verticessuch that 2 < clique number =chromatic number=k < n ,then the graph can be perfectly folded to a clique of order k.

The Ramsey number of loose cycles versus cliques

AbstractLet be the Ramsey number of an ‐uniform loose cycle of length versus an ‐uniform clique of order . Kostochka et al. showed that for each fixed , the order of magnitude of is up to a polylogarithmic factor in . They conjectured that for each we have . We prove that , and more generally for every that . We also prove that for every and , if is odd, which improves upon the result of Collier‐Cartaino et al. who proved that for every and we have .