In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\sf smallest defining set} (of vertex coloring) and its cardinality, the {\sf defining number}, is denoted by $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al \cite{rkgraph} proved that, for a given $k$ and for all $n \geq 3k$, if $r \geq 2(k-1)$ then $d(n, r, \Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $r\geq 2(k-1)$, $d(n, r, \Cchi=k)=k-1$.
In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$ directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of their smallest defining sets. We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except $v=5,15$ there exists a super-simple $(v,5,1)DD$. Also for these parameters, except possibly $v=11,91$, there exists a super-simple 2-$(v,5,1)$DD whose smallest defining sets have at least a half of the blocks.
In this note we show that for each Latin square $L$ of order $n\geq 2$, there exists a Latin square $L'\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Ω(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
In this paper, we show that for all $v\equiv 0,1$ (mod 5) and $v\geq 15$, there exists a super-simple $(v,5,2)$ directed design, also for these parameters there exists a super-simple $(v,5,2)$ directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes.
A $2-(v,k,λ)$ directed design (or simply a $2-(v,k,λ)DD$) is super-simple if its underlying $2-(v,k,2λ)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,λ)DD$ is simple if its underlying $2-(v,k,2λ)BIBD$ is simple, that is, it has no repeated blocks. A set of blocks which is a subset of a unique $2-(v,k,λ)DD$ is said to be a defining set of the directed design. A smallest defining set, is a defining set which has smallest cardinality. In this paper simultaneously we show that the necessary and sufficient condition for the existence of a sup…
If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\em defining set} for $M$. A {\em critical set} is a minimal defining set (the deletion of any entry results in more than one completion). We give a new classification of critical sets in $(0,1)$-matrices and apply this theory to $Λ_{2m}^m$, the set of $(0,1)$-matrices of dimensions $2m\times 2m$ with uniform row and column sum $m$. The smallest possible size for a defining set of a matrix in $Λ_{2m}^m$ is $m^2$ \cite{Cav}, and the infimum…
A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent $7$-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different $7$-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to $6\cdot 10^6$.
Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.
Perfect secret sharing scheme is a method of distribute a secret information $s$ among participants such that only predefined coalitions, called qualified subsets of the participants can recover the secret, whereas any other coalitions, the unqualified subsets cannot determine anything about the secret. The most important property is the efficiency of the system, which is measured by the information ratio. It can be shown that for graphs the information ratio is at most $(δ+1)/2$ where $δ$ is the maximal degree of the graph. Blundo et al. constructed a family of $δ$-regular graphs with informa…
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a …
By definition, the sharp packing index $\ind_P^\sharp(A)$ of a subset $A$ of an abelian group $G$ is the smallest cardinal $κ$ such that for any subset $B\subset G$ of size $|B|\geκ$ the family $\{b+A:b\in B\}$ is not disjoint. We prove that an infinite Abelian group $G$ contains a subset $A$ with given index $\ind_P^\sharp(A)=κ$ if and only if one of the following conditions holds: (1) $2\le κ\le|G|^+$ and $k\notin \{3,4\}$; (2) $κ=3$ and $G$ is not isomorphic to $\oplus_{i\in I} \mathbb{Z}_3$; (3) $κ=4$ and $G$ is not isomorphic to $\oplus_{i\in I} \mathbb{Z}_2$ or to $\mathbb{Z}_4\oplus(\op…
The construction of large-scale, low-latency networks becomes difficult as the number of nodes increases. In general, the way to construct a theoretically optimal solution is unknown. However, it is known that some methods can construct suboptimal networks with low-latency. One such method is to construct large-scale networks from optimal or suboptimal small networks, using the product of graphs. There are two major advantages to this method. One is that we can reuse small, already known networks to construct large-scale networks. The other is that the networks obtained by this method have gra…
Let k, r, s in the natural numbers where r \geq s \geq 2. Define f(s,r,k) to be the smallest positive integer n such that for every coloring of the integers in [1,n] there exist subsets S_1 and S_2 such that: (a) S_1 and S_2 are monochromatic (but not necessarily of the same color), (b) |S_1| = s, |S_2| = r, (c)max(S_1) < min(S_2), and (d) diam(S_1) \leq diam(S_2). We prove that the theorems defining f(s,r,2) and f(s,r,3) admit a partial generalization in the sense of the Erdos-Ginzburg-Ziv theorem. This work begins the off-diagonal case of the results of Bialostocki, Erdos, and Lefmann.
We discuss the amount of physical resources required to construct a given graph, where vertices are added sequentially. We naturally identify information -- distinct into instructions and memory -- and randomness as resources. Not surprisingly, we show that, in this framework, threshold graphs are the simplest possible graphs, since the construction of threshold graphs requires a single bit of instructions for each vertex and no use of memory. Large instructions without memory do not bring any advantage. With one bit of instructions and one bit of memory for each vertex, we can construct a fam…
Let X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is X. This quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the…
A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a $\varrho$-saturating set of $\text{PG}(N,q)$ has size at least $c\cdot\varrho\,q^\frac{N-\varrho}{\varrho+1}$, with $c>\frac{1}{3}$ a constant. Our main result is the discovery of a $\varrho$-saturating set of size roughly $\frac{(\varrho+1)(\varrho+2)}{2}q^\frac{N-\varrho}{\varrho+1}$ if $q=(q')^{\varrho+1}$, with $q'$ an arbitrary prime power. The existence of such…
We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a resul…
Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination number $γ_{md}(G)$ of the graph is the size of the smallest mixed dominating set of $G$. We present an explicit method for constructing optimal mixed dominating sets in Petersen graphs $P(n, k)$ for $k \in \{1, 2\}$. Our method also provides a new upper bound for other Petersen graphs.
