All posts under tag "Discrete Mathematics"
The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes
Community structure is an important area of research. It has received a considerable attention from the scientific community. Despite its importance, one of the key problems in locating information about community detection is the diverse spread of related articles across various disciplines. To the
In a graph $G$ of maximum degree $Delta$ let $gamma$ denote the largest fraction of edges that can be $Delta$ edge-coloured. Albertson and Haas showed that $gamma geq 13/15$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a grap
This paper presents an approach for the development of a number theoretic discrete Hilbert transform. The forward transformation has been applied by taking the odd reciprocals that occur in the DHT matrix with respect to a power of 2. Specifically, the expression for a 16-point transform is provided
We consider the Seidel complementation on ($P_5, bar{P_5}, Bull)$-free graphs
Let $mathcal {T}^{Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $Delta$. Suppose that every tree in $mathcal {T}^{Delta}_n$ is equally likely. We show that the number of vertices of degree $j$ in $mathcal {T}^{Delta}_n$ is asymptoticall
We investigate the complexity of approximately counting stable matchings in the $k$-attribute model, where the preference lists are determined by dot products of 'preference vectors' with 'attribute vectors', or by Euclidean distances between 'preference points' and 'attribute points'. Irving and Le
The notion of $mathcal{H}$-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of $mathcal{H}$-matroids by the greedy algorithm. In this note, we give a characterization of some $mathcal{H}$-matroids by
In a well-known paper[ARV], Arora, Rao and Vazirani obtained an O(sqrt(log n)) approximation to the Balanced Separator problem and Uniform Sparsest Cut. At the heart of their result is a geometric statement about sets of points that satisfy triangle inequalities, which also underlies subsequent work
Given n >= 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.
The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $tgeq 1$.
In this paper we propose a generic algorithm to calculate the rotation parameters of CORDIC angles required for the Discrete Cosine Transform algorithm (DCT). This leads us to increase the precision of calculation meeting any accuracy.Our contribution is to use this decomposition in CORDIC based DCT
Here the Integral Value Transformations (IVTs) are considered to be Discrete Dynamical System map in the spacemathbb{N}_(0). In this paper, the dynamics of IVTs is deciphered through the light of Topological Dynamics.
We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.
A set S is independent in a graph G if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=
A team of $r$ {it revolutionaries} and a team of $s$ {it spies} play a game on a graph $G$. Initially, revolutionaries and then spies take positions at vertices. In each subsequent round, each revolutionary may move to an adjacent vertex or not move, and then each spy has the same option. The revolu
The iterated Johnson bound is the best known upper bound on a size of an error-correcting code in the Grassmannian $mathcal{G}_q(n,k)$. The iterated Sch'{o}nheim bound is the best known lower bound on the size of a covering code in $mathcal{G}_q(n,k)$. We use probabilistic methods to prove that both
In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.
A Sierpinski number is an odd positive integer, k, such that no positive integer of the form k * 2^n + 1 is prime. Similar to a Sierpinski number, a Riesel number is an odd positive integer, k, such that no positive integer of the form k * 2^n + 1 is prime. A cover for such a k is a finite list of p
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and subtraction throughout. Presentation of material is non-theoretic
To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is 'fine' if there
A set S is independent if no two vertices from S are adjacent. In this paper we prove that if F is a collection of maximum independent sets of a graph, then there is a matching from S-{intersection of all members of F} into {union of all members of F}-S, for every independent set S. Based on this fi
Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ``planted'' solution; thus, we obtain the first rigorous result on BP for graph coloring in the case of a complex gr
Iteration semirings are Conway semirings satisfying Conway's group identities. We show that the semirings $N^{rat}llangle Sigma^* rrangle$ of rational power series with coefficients in the semiring $N$ of natural numbers are the free partial iteration semirings. Moreover, we characterize the semirin
We give an algorithm to route a multicommodity flow in a planar graph $G$ with congestion $O(log k)$, where $k$ is the maximum number of terminals on the boundary of a face, when each demand edge lie on a face of $G$. We also show that our specific method cannot achieve a substantially better conges
Let alpha(G) be the cardinality of a independence set of maximum size in the graph G, while mu(G) is the size of a maximum matching. G is a Konig--Egervary graph if its order equals alpha(G) + mu(G). The set core(G) is the intersection of all maximum independent sets of G (Levit & Mandrescu, 2002).
The problem of estimating the proportion of satisfiable instances of a given CSP (constraint satisfaction problem) can be tackled through weighting. It consists in putting onto each solution a non-negative real value based on its neighborhood in a way that the total weight is at least 1 for each sat
Given a graph $G$, an {em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoid
We present an $O^*(1.0919^n)$-time algorithm for finding a maximum independent set in an $n$-vertex graph with degree bounded by 3, which improves the previously known algorithm of running time $O^*(1.0977^n)$ by Bourgeois, Escoffier and Paschos [IWPEC 2008]. We also present an $O^*(1.1923^k)$-time
A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an e
Problems of existence, construction and estimation of parameters of interval colorings of complete k-partite graphs K_{n}^{k} are investigated.
We focus on the average-case analysis: A function w : V -> Z+ is given which defines the likelihood for a node to be the one marked, and we want the strategy that minimizes the expected number of queries. Prior to this paper, very little was known about this natural question and the complexity of th
A Conway semiring is a semiring $S$ equipped with a unary operation $^*:S to S$, always called 'star', satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as $N$ or $N^{rat}llangle Sigma^*
A partial monoid $P$ is a set with a partial multiplication $times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $star$ is simulated by a string rewriting system on $P^*$ that consists in evaluating
We show that any submodular minimization (SM) problem defined on a linear constraint set with constraints having up to two variables per inequality, are 2-approximable in polynomial time. If the constraints are monotone (the two variables appear with opposite sign coefficients) then the problems of
Sequences set is a mathematical model used in many applications. As the number of the sequences becomes larger, single sequence set model is not appropriate for the rapidly increasing problem sizes. For example, more and more text processing applications separate a single big text file into multiple
We derive the probability that a randomly chosen NL-node over $S$ gets localized as a function of a variety of parameters. Then, we derive the probability that the whole network of NL-nodes over $S$ gets localized. In connection with the asymptotic thresholds, we show the presence of asymptotic thre
In this paper the theory of Carry Value Transformation (CVT) is designed and developed on a pair of n-bit strings and is used to produce many interesting patterns. One of them is found to be a self-similar fractal whose dimension is same as the dimension of the Sierpinski triangle. Different constru
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {em weakly robust} polynomial time approx
We study the time and space complexity of matrix permanents over rings and semirings.
A set S is independent in a graph G if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. A connected graph hav
We establish a limit formula for the median of the distance between two leaves in a fully resolved unrooted phylogenetic tree with n leaves. More precisely, we prove that this median is equal, in the limit, to the square root of 4*ln(2)*n.
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small
Let $S_n^lambda$ be the set of all permutations over the multiset ${overbrace{1,...,1}^{lambda},...,overbrace{m,...,m}^lambda}$ where $n=mlambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^lambda$ in which every two elements have distance at least $d$. FPAs have
In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distrib
For three positive integers ai, aj, ak pairwise coprime, we present an algorithm that find the least multiple of ai that is a positive linear combination of aj, ak. The average running time of this algorithm is O(1). Using this algorithm and the chinese remainder theorem leads to a direct computatio
For integers r and k > 0(k>r),a conditional (k, r)-coloring of a graph G is a proper k-coloring of G such that every vertex v of G has at least min{r,d(v)} differently colored neighbors, where d(v) is the degree of v. In this note, for different values of r we obtain the conditional chromatic number
Let G=(V,E). A set S is independent if no two vertices from S are adjacent. 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}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G
Ziv-Lempel and Crochemore factorization are two kinds of factorizations of words related to text processing. In this paper, we find these factorizations for standard epiesturmian words. Thus the previously known c-factorization of standard Sturmian words is provided as a special case. Moreover, the
In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence of the effec
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