Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch and Radcliffe raised a natural variant of this problem where the number of edges is fixed instead of the number of vertices. In this paper, we determine the maximum number of cliques of a fixed order in a graph with fixed number of edges and bounded maximum degree, resolving a conjecture by Kirsch and Radcliffe. We also give a complete characterization of the extremal graphs.
In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo's recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.
Let $k_r(n,δ)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $δ$. We evaluate $k_r(n,δ)$ for $δ\leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture to give all extremal graphs (subject to certain conditions on $n$, $δ$ and $r$).
A \emph{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 $8(n-2)$.
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.
A complete subgraph of any simple graph $G$ on $k$ vertices is called a $k$-\emph{clique} of $G$. In this paper, we first introduce the concept of the value of a $k$-clique ($k>1$) as an extension of the idea of the degree of a given vertex. Then, we obtain the generalized version of handshaking lemma which we call it clique handshaking lemma. The well-known classical result of Mantel states that the maximum number of edges in the class of triangle-free graphs with $n$ vertices is equal to $\frac{n^{2}}{4}$. Our main goal here is to find an extension of the above result for the class of $K_{ω+…
For a $k$-graph $\mathcal{F}\subset \binom{[n]}{k}$, the clique number of $\mathcal{F}$ is defined to be the maximum size of a subset $Q$ of $[n]$ with $\binom{Q}{k}\subset \mathcal{F}$. In the present paper, we determine the maximum number of edges in a $k$-graph on $[n]$ with matching number at most $s$ and clique number at least $q$ for $n\geq 8k^2s$ and for $q \geq (s+1)k-l$, $n\leq (s+1)k+s/(3k)-l$. Two special cases that $q=(s+1)k-2$ and $k=2$ are solved completely.
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.
The Erdős--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of…
We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $χ$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $χ$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $χ$-bounded.
Erdos asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph of order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) when (k,l) is (3,4), (3,5), (3,6), (3,7), (4,3), (5,3), (6,3), and (7,3). Independently, Das, Huang, Ma, Naves, and Sudakov did the cases (k,l)=(3,4) and (4,3).
The Open University, School of Mathematics and Statistics（开放大学数学与统计学院）；Czech Technical University, Department of Theoretical Computer Science（捷克理工大学理论计算机科学系）；University of Florida, Department of Mathematics（佛罗里达大学数学系）
This paper studies the following question: Given a surface $Σ$ and an integer $n$, what is the maximum number of cliques in an $n$-vertex graph embeddable in $Σ$? We characterise the extremal graphs for this question, and prove that the answer is between $8(n-ω)+2^ω$ and $8n+{3/2} 2^ω+o(2^ω)$, where $ω$ is the maximum integer such that the complete graph $K_ω$ embeds in $Σ$. For the surfaces $\mathbb{S}_0$, $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{N}_3$ and $\mathbb{N}_4$ we establish an exact answer.
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.
If a graph has no induced subgraph isomorphic to $H_1$ or $H_2$ then it is said to be ($H_1,H_2$)-free. Dabrowski and Paulusma found 13 open cases for the question whether the clique-width of ($H_1,H_2$)-free graphs is bounded. One of them is the class of ($S_{1,2,2}$,triangle)-free graphs. In this paper we show that these graphs have bounded clique-width. Thus, also ($P_1+2P_2$,triangle)-free graphs have bounded clique-width which solves another open problem of Dabrowski and Paulusma. Meanwhile we were informed by Paulusma that in December 2015, Dabrowski, Dross and Paulusma showed that ($S_{…
There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus 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 explore evidences that the Lagrangian of a 3-uniform hypergraph is related to the order of its maximum cliques when the number of edges of the hypergraph is in certain range. In particular, we present some results about a conjectu…
There are many methods to find a maximum (or maximal) clique in large networks. Due to the nature of combinatorics, computation becomes exponentially expensive as the number of vertices in a graph increases. Thus, there is a need for efficient algorithms to find a maximum clique. In this paper, we present a graph reduction method that significantly reduces the order of a graph, and so enables the identification of a maximum clique in graphs of large order, that would otherwise be computational infeasible to find the maximum. We find bounds of the maximum (or maximal) clique using this reductio…
We present a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. This result provides a negative answer to the question asked by Daligault, Rao and Thomassé in ("Well-quasi-order of relabel functions", Order, 27(3):301--315, 2010).
The clique-width is known to be unbounded in the class of unit interval graphs. In this paper, we show that this is a minimal hereditary class of unbounded clique-width, i.e., in every hereditary subclass of unit interval graphs the clique-width is bounded by a constant.
This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.
We characterize the connected graphs of given order $n$ and given independence number $α$ that maximize the number of maximum independent sets. For $3\leq α\leq n/2$, there is a unique such graph that arises from the disjoint union of $α$ cliques of orders $\left\lceil\frac{n}α\right\rceil$ and $\left\lfloor\frac{n}α\right\rfloor$, by selecting a vertex $x$ in a largest clique and adding an edge between $x$ and a vertex in each of the remaining $α-1$ cliques. Our result confirms a conjecture of Derikvand and Oboudi [On the number of maximum independent sets of graphs, Transactions on Combinato…
A class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is chi-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is chi-bounded.
Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1…
We prove that every graph $G$ for which $ω(G) \geq 3/4(Δ(G) + 1)$, has an independent set $I$ such that $ω(G - I) < ω(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $ω(G) < 3/4(Δ(G) + 1)$ and hence also $χ(G) > \lceil 7/6ω(G) \rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $χ(H) \leq \max{\lceil 7/6ω(H) \rceil, \lceil \frac{ω(H) + Δ(H) + 1}{2}\rceil}$, then we also have $χ(G) \leq \lceil \frac{ω(G) + Δ(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.
It was recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $ω\geq \frac 23(Δ+1)$ unless the graph is the strong product of $K_{ω/2}$ and an odd hole. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.
The $kK_{r+1}$ is the union of $k$ disjoint copies of $(r+1)$-clique. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then $K_{k-1}\vee T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper, we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We prove that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to $K_{k-1}\vee T_{n…
Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hatβ}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hatβ}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hatβ}_t(G)$ including the connections to the greatest reduced average dens…
How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $2^{t+\log^2 t}n$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$.…
Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with omega > (2/3)(Delta+1). This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size.
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class $B_k$-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most $k$ bends. Epstein et al. showed in 2013 that computing a maximum clique in $B_1$-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at least $4$, the class contains $2$-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for $B_2$ and $B_3$-EP…
Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of $GP(q,d)$ is tight if and only if $d \mid (\sqrt{q}+1)$, which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickel…
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of $H$-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs $H$ for which the class of $H$-free split graphs has bounded clique-width.
Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determ…
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex $P_4$-extensions $H$ for which the class of $H$-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of $H$-free chordal graphs of bounde…
We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and more generally, of classes of bounded genus. To obtain this result we develop a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future.
