For given graph $H$, the independence number $α(H)$ of $H$, is the size of the maximum independent set of $V(H)$. Finding the maximum independent set in a graph is a NP-hard problem. Another version of the independence number is defined as the size of the maximum induced forest of $H$, and called the forest number of $H$, and denoted by $f(H)$. Finding $f(H)$ is also a NP-hard problem. Suppose that $H=(V(H),E(H))$ be a graph, and $\G$ be a family of graphs, a graph $H$ has a $\G$-free $k$-coloring if there exists a decomposition of $V(H)$ into sets $V_i$, $i-1,2,\ldots,k$, so that $G\nsubseteq H[V_i]$ for each $i$, and $G\in\G$. $S\subseteq V(H)$ is $G$-free, where the subgraph of $H$ induced by $S$, be $G$-free, i.e. it contains no copy of $G$. Finding a maximum subset of $H$, so that $H[S]$ be a $G$-free graph is a very hard problem as well. In this paper, we study the generalized version of the independence number of a graph. Also giving some bounds about the size of the maximum $G$-free subset of graphs is another purpose of this article.
There are two known families of maximum scattered $\mathbb{F}_q$-linear sets in $PG(1,q^t)$: the linear sets of pseudoregulus type and for $t\geq 4$ the scattered linear sets found by Lunardon and Polverino. For $t=4$ we show that these are the only maximum scattered $\mathbb{F}_q$-linear sets and we describe the orbits of these linear sets under the groups $PGL(2,q^4)$ and $PΓL(2,q^4)$.
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…
In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $ψ(G)$, is the cardinality of a maximum dissociation set of $G$. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and…
Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $γ$ has at most $2^γ$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $γ$ and $2.0598^γ$ minimum dominating sets. We show that every forest with domination number $γ$ has at most $2.4606^γ$ minimum dominating sets, and that every tree with independence number $α$ has at most $2^{α-1}+1$ maximum independent sets.
Given a connected graph with domination (or total domination) number γ>=2, we ask for the maximum number m_γand m_{γ,T} of dominating and total dominating sets of size γ. An exact answer is provided for γ=2and lower bounds are given for γ>=3.
A subset of vertices $F$ in a graph $G$ is called a \emph{dissociation set} if the induced subgraph $G[F]$ of $G$ has maximum degree at most 1. A \emph{maximal dissociation set} of $G$ is a dissociation set which is not a proper subset of any other dissociation sets. A \emph{maximum dissociation set} is a dissociation set of maximum size. We show that every graph of order $n$ has at most $10^{\frac{n}{5}}$ maximal dissociation sets, and that every triangle-free graph of order $n$ has at most $6^{\frac{n}{4}}$ maximal dissociation sets. We also characterize the extremal graphs on which these up…
In a graph G, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of maximum dissociation set problem in various subclasses of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and to no maximum dissociation set of a tree. We…
We give a very short and simple proof of Zykov's generalization of Turán's theorem, which implies that the number of maximum independent sets of a graph of order $n$ and independence number $α$ with $α<n$ is at most $\left\lceil\frac{n}α\right\rceil^{n\,{\rm mod}\,α} \left\lfloor\frac{n}α\right\rfloor^{α-(n\,{\rm mod}\,α)}$. Generalizing a result of Zito, we show that the number of maximum independent sets of a tree of order $n$ and independence number $α$ is at most $2^{n-α-1}+1$, if $2α=n$, and, $2^{n-α-1}$, if $2α>n$, and we also characterize the extremal graphs. Finally, we show that the n…
We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number γ, of dominating sets of size γ. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all sets of size γ. At the same time, we use a modified adjacency matrix to obtain lower bounds on the number of sets of a given size that do not dominate a graph on n vertices
A subset of vertices in a graph $G$ is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1 and the subset has maximum cardinality. The dissociation number of $G$, denoted by $ψ(G)$, is the cardinality of a maximum dissociation set. A subcubic tree is a tree of maximum degree at most 3. In this paper, we give the lower and upper bounds on the dissociation number in a subcubic tree of order $n$ and show that the number of maximum dissociation sets of a subcubic tree of order $n$ and dissociation number $ψ$ is at most $1.466^{4n-5ψ+2}$.
We determine the shape of all sum-free sets in $\{1,2,\ldots,n\}^2$ of size close to the maximum $\frac{3}{5}n^2$, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe $\frac{4}{5}n-o(n)\le x+y\le\frac{8}{5}n+ o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $A\subseteq \{1,2,\ldots,n\}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.
Linear set in projective spaces over finite fields plays central roles in the study of blocking sets, semifields, rank-metric codes and etc. A linear set with the largest possible cardinality and the maximum rank is called maximum scattered. Despite two decades of study, there are only a few number of known maximum scattered linear sets in projective lines, including the family constructed by Csajbók, Marino, Polverino and Zanella 2018, and the family constructed by Csajbók, Marino, Zullo 2018 (also Marino, Montanucci, and Zullo 2020). This paper aims to solve the equivalence problem of the li…
A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito [J. Graph Theory {\bf 15} (1991) 207--221] proved that the maximum number of maximum independent sets of a tree of order $n$ is $2^{\frac{n-3}{2}}$ if $n$ is odd, and $2^{\frac{n-2}{2}}+1$ if $n$ is even and also characterized all extremal trees with the most maximum independent sets, which solved a …
In [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces of $V(2,q^n)$ have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the $\mathbb{F}_q$-linear sets presented in [23] and in [5], for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,\,0 \pmod 5$, we present new examples of maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^6)$, ari…
The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ and odd $q$. In particular, the relevant family contains …
A two-intersection set with parameters $(j;α,β)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $α$ or $β$ points. In this paper, we show the existence of a two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block design obtained from translations of the set of nonzero squares in the finite field of order $q=4m^2+1$. As an application, we give a construction of conference matrices with maximum excess based on the two-intersection sets.
The rank of a scattered $\mathbb{F}_q$-linear set of $\mathrm{PG}(r-1,q^n)$, $rn$ even, is at most $rn/2$ as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of $r$, $n$, $q$ ($rn$ even) for scattered $\mathbb{F}_q$-linear sets of rank $rn/2$. In this paper we prove that the bound $rn/2$ is sharp also in the remaining open cases. Recently Sheekey proved that scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^n)$ of maximum rank $n$ yield $\mathbb{F}_q$-linear MRD-codes with dimension $2n$ and minimum distance $n-1$.…
We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis$_\bigtriangleup(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis$_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis$_{\bigtr…
We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n/4$, and no set of less than $\sqrt{n}/2$ vertices intersects all its maximum independent sets. This is motivated by a question of Bollobás, Erdős and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph $G$ on $n$ vertices with independence number $(1/4+\eps)n$, the average independence number of an induced subgraph of $G$ on a uniform random subset of the vertices is at most $(1/4+\eps…
The (n,k)-star graph is a very important computer modelling. The independent number and chromatic number of a graph are two important parameters in graph theory. However, we did not know the values of this two parameters of the (n,k)-star graph since it was proposed. In [18], Wei et. al. declared that they determined the independent number of the (n,k)-star graph, unfortunately their proof is wrong. This paper generalize their result and present a maximum independent sets partition of (n,k)-star graph. From that we can immediately deduce the exact value of the independent number and chromatic …
Erdős and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these lines. A $k$-dominating independent set is an independent set $D$ such that every vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$ denote the maximum number of $k$-dominating independent sets in a graph on $n$ vertices, and let $ζ_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$. Nagy initiated the study of $mi_k(n)$. In this artic…
In this paper we study a family of scattered $\F_q$--linear sets of rank $tn$ of the projective space $PG(2n-1,q^t)$ ($n \geq 1$, $t\geq 3$), called of {\it pseudoregulus type}, generalizing results contained in [G. Marino, O. Polverino, R. Trombetti: On ${\mathbb F}_q$--linear sets of $\PG(3,q^3)$ and semifields, {\em J. Combin. Theory Ser. A}, {\bf 114} (2007), 769--788] and in [M. Lavrauw, G. Van de Voorde: Scattered linear sets and pseudoreguli, preprint]. As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twis…
In this paper, we investigate a problem concerning quartets, which are a particular type of tree on four leaves. Loosely speaking, a set of quartets is said to be `definitive' if it completely encapsulates the structure of some larger tree, and `minimal' if it contains no redundant information. Here, we address the question of how large a minimal definitive quartet set on n leaves can be, showing that the maximum size is at least 2n-8 for all n>3. This is an enjoyable problem to work on, and we present a pretty construction, which employs symmetry.
Explicit constructions of infinite families of scattered ${\mathbb F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even, are provided. When $q=2$ and $r$ is odd, these linear sets correspond to complete caps in $AG(r,2^t)$ fixed by a translation group of size $2^{\frac{rt}2}$. The doubling construction applied to such caps gives complete caps in $AG(r+1,2^t)$ of size $2^{\frac{rt}2+1}$. For Galois spaces of even dimension greater than $2$ and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a …
A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.
In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $Γ$ and a canonical subgeometry $Σ$ in $\mathrm{PG}(n-1,q^n)$, with $Γ\capΣ=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $Σ$ from the vertex $Γ$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the project…
The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs B(d,3) and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.
A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a …
The maximum scattered linear sets in $PG(1,q^n)$ have been completely classified for $n \le 4$ by Csajbók-Zanella and Lavrauw-Van de Voorde. Here a wide class of linear sets in $PG(1,q^5)$ is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in $PG(1,q^5)$ are exhibited.
For a permutation group $G$ acting on a set $V$, a subset $I$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in I$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|I|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $I$ runs over all intersecting sets in $G$. If $G_v$ is the largest intersecting set in $G$ then $G$ is said to have the Erdős-Ko-Rado (EKR)-property, and moreover, $G$ has the strict-EKR-property if every intersecting set of maximum siz…
An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in early 70's. The class of graphs for which every induced subgraph admits two disjoint maximal independent sets was characterized in (Shaudt, 2015). It is known that deciding whether a graph has two disjoint maximal independent sets is a NP-complete problem (Henning et al., 2009). In this paper, we are focused on finding conditions ensuring the existence of two …
A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. It is known that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs whose families of local maximum stable sets form greedoids have been analyzed as well. A unicycle graph owns only one cycle. In this paper we characterize the unicycle graphs whose families of local maximum stable sets form greedoids.
For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a \textit{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements ${\mathbf{a}}_1, \ldots, {\mathbf{a}}_k$ in $S$ satisfying ${\mathbf{a}}_1+\cdots+{\mathbf{a}}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a {$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$. This is the fir…
We prove constructively that the maximum possible number of minimal connected dominating sets in a connected undirected graph of order $n$ is in $Ω(1.489^n)$. This improves the previously known lower bound of $Ω(1.4422^n)$ and reduces the gap between lower and upper bounds for input-sensitive enumeration of minimal connected dominating sets in general graphs as well as some special graph classes.
The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. The maximum independent sets in the Doob graphs are analogs of the distance-$2$ MDS codes in the Hamming graphs. We prove that the logarithm of the number of the maximum independent sets in $D(m,n)$ grows as $2^{2m+n-1}(1+o(1))$. The main tool for the upper estimation is constructing an injective map from the class of maximum independent sets in $D(m,n)$ to the class of distance-$2$ MDS codes in $H(2m+n,4)$.
Paul Erdős and László Lovász proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the problem of determining or estimating the largest number $N(k)$ of the points in such a family. They also proved by means of an example that $N(k)\geq2k-2+\frac{1}{2}\binom{2k-2}{k-1}$. Much later, Zsolt Tuza proved that the bound is best possible up to a multiplicative constant by showing that asymptotically $N(k)$ is at most $4$ times this lower bound. In this paper we red…
We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters $(6,6,q;5)$.
The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graph- s, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching n…
We investigate the relationship between two constructions of maximal comma-free codes described respectively by Eastman and by Scholtz and the notions of Hall sets and Lazard sets introduced in connection with factorizations of free monoids and bases of free Lie algebras.
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the only 12-point three-distance set in $\mathbb{R}^3$ up to isomorphism. In this paper, we prove the uniqueness of 12-point three-distance sets in $\mathbb{R}^3$.
Let G be a simple graph with vertex set V(G). A set S is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set in G} (Zhang; 1990). Let Omega(G) denote the family of all maximum independent sets. Let us say that a family Gamma of independent sets is a Konig-Egervary collection if |Unio…
Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,...,n\}$ and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$ the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph $A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this paper, we intensively investigate the properties of $A\T_n^q$ including connectedness, diameter, independence number, clique number, chromatic number and the maximum-size independent sets of $A\T_n^q.$ By using the result on the maximum-size independent sets $A\T_n^q$, we completely determine the …
Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kah…
