Vladimir Nikiforov
We prove that for all $r\geq2$ and c>0, every graph of order n with at least cn^{r} cliques of order r contains a complete r-partite graph with each part of size $\lfloor c^{r}\log n \rfloor.$ This result implies a concise form of the Erdős-Stone theorem.
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We give a minimum degree condition sufficent to ensure the existence of a fractional $K_r$-decomposition in a balanced $r$-partite graph (subject to some further simple necessary conditions). This generalises the non-partite problem studied recently by Barber, Lo, Kühn, Osthus and the author, and the $3$-partite fractional $K_3$-decomposition problem studied recently by Dukes. Combining our result with recent work by Barber, Kühn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional) $K_r$-decomposition in a balanced $r$-partite g…
We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex hulls intersect in a common point. This is achieved by translating the problem to $k$-partite graphs and using a new algorithm to verify that those graphs do not have a $k$-clique.
We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964.
On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs
本研究针对不含固定完全r部图的n顶点图中r团的最大数量问题，利用图移除引理将原有的Turán数上界从多项式阶改进为低阶多项式阶，并给出了若干下界构造，推进了极值图论中相关计数问题的研究。
For two graphs $G$ and $H$, the Turán number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Turán numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this paper we will determine the Turán numbers $ex(K_{a_1,\ldots,a_r},kK_r)$ for all $r\geq 3$ and $k\geq 1$.
In this note, we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay. It is proved that if $G$ is an $r$-partite Cohen-Macaulay graph which is covered by some disjoint cliques of size $r$, then the clique cover is unique.
In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.
The anti-Ramsey numbers of cliques in complete multi-partite graphs
本文研究了完全多部图中团 $K_k$ 的反拉姆齐数，并确定了其在完全 $k$-部图及 $r\geq k$ 的平衡完全 $r$-部图中的具体数值。
Largest $2$-regular Subgraphs in complete $S$-partite Graphs
研究完全S-部图中最大2-正则子图问题，提出通过整数线性规划求解，并证明该问题可在O(|V(S)|^3)时间内高效解决，同时通过模拟验证随机S-部图中存在与完整图相同阶数的最大2-正则子图。
Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translate…
We provide a new family of $K_k$-free pseudorandom graphs with edge density $Θ(n^{-1/(k-1)})$, matching a recent construction due to Bishnoi, Ihringer and Pepe. As in the former result, the idea is to use large subgraphs of polarity graphs, which are defined over a finite field $\mathbb{F}_q$. While their construction required $q$ to be odd, we will give the first construction with $q$ even.
We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the the r-partite Turan graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.
Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$ vertex-disjoint triangles. For any $r> t\ge 3$, we give a conjecture on the maximum number of edges of an $r$-partite graph that does not contain $k$ vertex-disjoint cliques $K_t$.
For two $s$-uniform hypergraphs $H$ and $F$, the Turán number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq \cdots\leq n_r$. De Silva, Heysse and Young determined $ex_2(K_{n_1, \ldots, n_r}, kK_2)$ and De Silva, Heysse, Kapilow, Schenfisch and Young determined $ex_2(K_{n_1, \ldots, n_r},kK_r)$. In this paper, as a generalization of these results, we consider three Turán-type problems for $k$ disjoint cliques in $r$-partite $s$-uniform hypergraphs. First, we consider a multi-…
We look at structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph $H$ and an oriented graph $G$, let $f_H(G)$ be the maximum number of pairwise disjoint copies of $H$ that can be found in {\em all} feedback arc sets of $G$. In particular, to make $G$ acyclic, one must remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Most intriguing is the case where $H$ is a $k$-clique, where the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we determine $f_k(G)$ precisely for al…
In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.
We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $ε:=ε(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which all degrees are at least $(1-ε)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k-1)$-colorable. We also consider several extensions for the case of a general forbidden graph $H$ of a given …
Clique supersaturation under a chromatic constraint below the Turán threshold
本文针对Turán阈值以下非p部图的场景，证明了满足边数条件的图中K_{p+1}副本数的精确团计数界，其构造达到该界，为Brouwer阈值提供了团计数类似物。
Fractional clique decompositions of dense balanced multipartite graphs
研究密集平衡多部图的分数 clique 分解条件，提出新的度数阈值，适用于多于 s 部的多部图。
Hypergraphs without complete partite subgraphs
本文研究了不含特定完全部分图的超图的扎兰基数，通过改进的方法证明了当t大于某个阈值时，扎兰基数的上界为n^{r-1/s-o(1)}，并给出了具体例子中的最优指数。
Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density $d^k_l$, such that every $\ell$-partite graph whose parts have pairwise edge density greater than $d^k_l$ contains a $K^k$. It turns out that $d^k_l=(k-2)/(k-1)$ for large enough l. We also describe the structure of the extremal graphs. For the case of triangles we show that $d^3_{13}=1/2$, disproving a conjecture by Bondy, Shen, Thomasse and Thomassen.
The number of cliques in hypergraphs with forbidden subgraphs
本文研究禁用完全 r 部超图的 $(r-1)$-一致超图中 r 团的最大数量，借助超图移除引理证明上界，并用超图 Turán 数给出下界。
A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number $scc(G)$ of a graph $G$, is defined as the smallest possible weight of a clique covering of $G$. Let $ K_t(d) $ denote the complete $ t $-partite graph with each part of size $d$. We prove that for any fixed $d \ge 2$, we have $$\lim_{t \rightarrow \infty} scc(K_t(d))= \frac{d}{2} t\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi.
The classical Hadwiger conjecture dating back to 1940's 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 $α(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/α(G)$. Results of Kuhn and Osthus and Krivelevich and Sudako…
A stability theorem for multi-partite graphs
本文提出多部图的稳定性定理，解决了关于不含顶点不相交克lique副本的多部图最大边数的猜想。
Maximizing subgraph density in graphs of bounded degree and clique number
本文渐近确定了在有界度数和团数的图中包含支配顶点的子图的最大密度，通过固定u-团数目的统一推广证明及局部化方法，得出了相关极值与渐近性质。
In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \leqslant (\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of…
In 1994, Cornelis Hoede and Xueliang Li introduced the clique polynomial of a graph. Also, a theorem for the edge subgraph expansion for clique polynomials. In this note we present a counter example for it and explain which case it could be a valid theorem.
This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number r, then the largest eigenvalue q(G) of the matrix Q=A+D satisfies q(G)<= 2(1-1/r)n. If G is a complete regular r-partite graph, then equality holds in the above inequality. This result confirms a conjecture of Hansen and Lucas.
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width …
If a graph has no induced subgraph isomorphic to any graph in a finite family $\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of $H$-free graphs has bounded clique-width if and only if $H$ is an induced subgraph of the 4-vertex path $P_4$. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs $H_1$ and $H_2$. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded cli…
We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various val…
The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent i…
Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques.
Localization of the clique spectral version of Zykov's theorem
本文研究图的团谱版本齐肯图定理的局部化问题，通过给出图的团张量的谱半径上界，扩展了齐肯图定理的谱版本。
On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs
本文通过旗帜代数方法，解决了禁止子图图中 $s$- clique 密度最小化的问题，并应用于解决 Baumann 和 Briggs 提出的若干问题。
Some recent results on graph eigenvalues are improved. In particular, among all graphs of given order with no cliques of order $(r+1)$ the $r$-partite Turan graph has maximal spectral radius.
The $(a,b)$-eye is the graph $I_{a,b} = K_{a+b}-K_b$ obtained by deleting the edges of a clique of size $b$ from a clique of size $a+b$. We show that for any $a,b \ge 2$ and $p \in (0,1)$, if we condition the random graph $G \sim G(n,p)$ on having no induced copy of $I_{a,b}$, then with high probability $G$ is close to an $a$-partite graph or the complement of a $(b-1)$-partite graph. Our proof uses the recently developed theory of hypergraph containers, and a stability result for an extremal problem with two weighted colours. We also apply the stability method to obtain an exact Turán-type re…
We look at several saturation problems in complete balanced blow-ups of graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and refer to a copy of $H$ in $H[n]$ as 'partite' if it has one vertex in each part of $H[n]$. We then ask how few edges a subgraph $G$ of $H[n]$ can have such that $G$ has no partite copy of $H$ but such that the addition of any new edge from $H[n]$ creates a partite $H$. When $H$ is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger. Our main result is to calculate this value for $H=K_4$ when $n$ is large. We also give exac…
The thickness of a graph G is the minimum number of planar subgraphs whose union is G. In this paper, we obtain the thickness of complete 3-partite graph K_1,n,n, K_2,n,n and complete 4-partite graph K_1,1,n,n.
We prove that the clique graph operator $k$ is divergent on a locally cyclic graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $δ(G)=6$ if and only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the maximal complete subgraphs of $G$ as vertices, and the edges are given by non-empty intersections. If all iterated clique graphs of $G$ are pairwise non-isomorphic, the graph $G$ is $k$-divergent; otherwise, it is $k$-convergent. To prove our claim, we explicitly construct the iterated clique graphs of those infinite locally cyclic graphs with $δ\geq6$ which induce simply c…
A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdös) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a "higher-dimensional" analogue of the notion of an induced $K_{2,2}$ which allows us to generalize their result to $k$-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a …
In 1975 Bollobás, Erd\H os, and Szemerédi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest minimum degree $δ(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szabó, and Szabó and Tardos in 2006. In this paper we investigate the $r>t+1$ case of the problem, which has remained dormant for over forty years. We resolve the problem exactly in the case when $r \equiv -1 \pmod{t}$, and up to an additive…
Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is well-known that such parameters are related to certain vertex coverings. Continuing along these lines, we investigate positive zero forcing within the context of certain clique coverings. A key object considered here is the compressed cliques graph. We study a number of properties associated with the compressed cliques graph, including: uniqueness, forbidden subgraphs, connections to Johnson graphs, and positive zero …
Maximum cliques in a graph without disjoint given subgraph
本文研究广义 Turán 数，确定了 $\ex(n,K_3,2C_5)$ 的精确值并刻画大 $n$ 时的唯一极图，同时给出 $\ex(n,K_r,(k+1)K_r)$ 的精确值，推广了已有结果。
We prove a conjecture of Andras Gyarfas, that for all k,t, every graph with clique number at most k and sufficiently large chromatic number has an odd hole of length at least t.
A graph $G$ is called $C_4$-free if it does not contain the cycle $C_4$ as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphs with $n$ vertices and average degree at least $cn$ contain a complete subgraph (clique) of size at least $c'n$ (with $c'= 0.1c^2n$). We prove here better bounds (${c^2n\over 2+c}$ in general and $(c-1/3)n$ when $ c \le 0.733$) from the stronger assumption that the $C_4$-free graphs have minimum degree at least $cn$. Our main result is a theorem for regular graphs, c…
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices associated by a simple graph $G$ as a generalization of the classical vertex-edge incidence matrix of $G$. Then, using these clique incidence matrices, we obtain two clique-counting identities that can be used for deriving two combinatorial formulas for the first and the second derivatives of clique polynomials. Finally, we conclude the paper with several open que…
Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden subgraphs is necessary in the hypothesis.
Finding cliques and dense subgraphs using edge queries
该研究针对Erdős–Rényi随机图中有限边查询下找大团的问题，改进了多轮查询场景下团规模系数的上界，同时拓展到稠密子图查找问题并证明了对应不可能性结果。
We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with nonnegative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope …
For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $χ(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $χ(H)-1$ partite subgrap…
On maximal cliques in the graph of simplex codes
本文研究单纯码在 Grassmann 图中形成的诱导子图，刻画其两类极大团，并证明第一类团可由单项式线性自同构相互转移，第二类团结构更复杂且大小可变。
We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length $3$ or $4$. The second problem is about the perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect it it induces a perfect graph.) The third problem is on antichains in the induced subgraph ordering. The fourth problem is on graphs in which the difference betw…
It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$ cliques suffice; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon.
Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal F}$ is a subgraph of $J+e$. We let $\text{ex}({\mathcal F},G)$ and $\text{sat}({\mathcal F},G)$ denote the maximum and minimum size of an ${\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\mathcal F}$ is a subgraph of $G$, then $\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|$. In this paper, for $k\ge 3$ and $n\ge 100…
An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering such that every edge of $H$ has nonempty intersection with each of the intervals and is contained in their union. Our main result implies that for each $α> k - 1$ and $d>0$, every $n$-vertex ordered $r$-graph with $d \,n^α$ edges has for some $m\leq n$ an $m$-vertex interval $k$-partite subgraph with $Ω(d\, m^α)$ edges. This is an extension to ordered $r$-g…
Let $\hom(G)$ denote the size of the largest clique or independent set of a graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$ with $\hom(G) = O(\log n)$ contains an induced subgraph with $Ω(n^{1/2})$ distinct degrees, and raised the question of deciding whether an analogous result holds for every $n$-vertex graph $G$ with $\hom(G) = O(n^ε)$, where $ε> 0$ is a fixed constant. Here, we answer their question in the affirmative and show that every graph $G$ on $n$ vertices contains an induced subgraph with $Ω((n/\hom(G))^{1/2})$ distinct degrees. We also prove a stronger …
Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set of neighbours of $v$ outside of $K$ is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs.
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