All posts under tag "Discrete Mathematics"
The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e.
The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $Deltageq12$ and $d=2$, the (2,1)-total labelling number $lambda_2^T(G)$ is at most $Delta+2$.
We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $textbf{A}$ of length $N gg B$. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of $hat{textbf{A}}$, and estimate thei
The complexity of dual basis of a type I optimal normal basis of $mathbb{F}_{q^n}$ over $mathbb{F}_{q}$ was determined to be $3n-3$ or $3n-2$ according as $q$ is even or odd, respectively, by Z.-X. Wan and K. Zhou in 2007. We give a new proof to this result by clearly deriving the dual of a type I o
We investigate the Boolean functions with essential arity gap 2. We use Full Conjunctive Normal Forms instead of Zhegalkin's polynomials, which allow us to simplify the proofs and to obtain several combinatorial results, concerning the Boolean functions with a given arity gap.
An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex v are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In
The walk distances in graphs are defined as the result of appropriate transformations of the $sum_{k=0}^infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic,
The independence number of a graph G, denoted by alpha(G), is the cardinality of an independent set of maximum size in G, while mu(G) is the size of a maximum matching in G, i.e., its matching number. G is a Konig-Egervary graph if its order equals alpha(G)+mu(G). In this paper we give a new charact
The Turan bound is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro-Wei inequality, which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We sh
We analyze the asymptotic behavior of sequences of random variables defined by an initial condition, a stationary and ergodic sequence of random matrices, and an induction formula involving multiplication is the so-called max-plus algebra. This type of recursive sequences are frequently used in appl
This paper is concerned with palindromes occurring in characteristic Sturmian words $c_alpha$ of slope $alpha$, where $alpha in (0,1)$ is an irrational. As $c_alpha$ is a uniformly recurrent infinite word, any (palindromic) factor of $c_alpha$ occurs infinitely many times in $c_alpha$ with bounded g
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as 'counting in two ways'. The principle of inclusion and exclusion is used for
In this paper, we prove that most of the boolean functions, $f : {-1,1}^n rightarrow {-1,1}$ satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy of a boolean function is at most a constant times the Influence of the fu
We design a 3/2 approximation algorithm for the Generalized Steiner Tree problem (GST) in metrics with distances 1 and 2. This is the first polynomial time approximation algorithm for a wide class of non-geometric metric GST instances with approximation factor below 2.
Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of non-treelike evolutionary events, like recombination, hybridization, or lateral gene transfer. In this paper, we present and study a new class of phylogenetic networks, called tree-child phylogeneti
In this work, a wide family of LFSR-based sequence generators, the so-called Clock-Controlled Shrinking Generators (CCSGs), has been analyzed and identified with a subset of linear Cellular Automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algor
We show that P2T - the problem of deciding whether the edge set of a simple graph can be partitioned into two trees or not - is NP-complete.
Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number $chi(G)$, for a
A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in cyclotomic rings and cyclotomic fields in that most arithmetic
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