We exhibit a set system S 0 such that any set system containing it has linear discrepancy at least 2. Let V be a nite set and S 2 V a system of subsets of V. The incidence matrix A of (V; S) has rows indexed by the sets of S and the columns indexed by the points of V. The column corresponding to a point v 2 V is denoted by a v. A coloring of V is a mapping : V ! f?1; +1g. The discrepancy of S, ...

LetH = (V, E) be a hypergraph and let A be the E−V incidence matrix. We callH box-Mengerian if the linear system Ax ≥ 1, x ≥ 0 is box-totally dual integral (box-TDI). As it is NP -hard in general to recognize box-Mengerian hypergraphs, a basic theme in combinatorial optimization is to identify such objects associated with various problems. In this paper we show that the so-called ESP (equitable...

We want to examine line-free subsets of projective spaces over GF(2) especially from an intersection-set point of view. In this paper, we describe incidence equations for such sets. For every n~>3, we construct a line-free subset (or cap), X,, of PG(n,2), which meets every hyperplane in at least 2" 3 points. We make use of the incidence equations and other methods to show that, remarkably, up t...

In this paper we will estimate the main parameters of some evaluation codes which are known as parameterized codes. We will find the length of these codes and we will give a formula for the dimension in terms of the Hilbert function associated to the vanishing ideal of the projective torus. Also we will find an upper bound for the minimum distance and, in some cases, we will give some lower bou...

A fuzzy graph is a symmetric binary fuzzy relation on a fuzzy subset. The concept of fuzzy sets and fuzzy relations was introduced by L.A.Zadeh in 1965cite{zl} and further studiedcite{ka}. It was Rosenfeldcite{ra} who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975. The concepts of fuzzy trees, blocks, bridges and cut nodes in fuzzy graph has been studi...

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k > 0 colors. Each different ruleset specifies that game’s coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, an...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...

A new variation of the coloring problem, μ-coloring, is defined in this paper. Given a graph G and a function μ, a μ-coloring is a coloring where each vertex v of G must receive a color at most μ(v). It is proved that μ-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. The notion of perfection is extended for μ-coloring, l...

Dvo?ák and Postle introduced the concept of DP-coloring to overcome some difficulties in list coloring. Sittitrai Nakprasit combined defective coloring define a new coloring—relaxed DP-coloring. For relaxed DP-coloring, Sribunhung et al. proved that planar graphs without 4- 7-cycles are DP-(0, 2, 2)-colorable. Li 4, 8-cycles or 9-cycles DP-(1, 1, 1)-colorable. Lu Zhu 5-cycles, 6-cycles, In this...