On the Weight Distribution of the Coset Leaders of Constacyclic Codes

Constacyclic codes with one and the same generator polynomial and distinct length are considered. We give a generalization of the previous result of the first author [4] for constacyclic codes. Suitable maps between vector spaces determined by the lengths of the codes are applied. It is proven that the weight distributions of the coset leaders don’t depend on the word length, but on generator polynomials only. In particular, we prove that every constacyclic code has the same weight distribution of the coset leaders as a suitable cyclic code

Partitions of graphs into small and large sets

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $\deg(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in B$, $\deg(u) \ge |B| - k - 1$. Moreover, we denote by $\varphi_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-small sets, and by $\Omega_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-large sets. In this paper, we will show tight connections between $k$-small sets, respectively $k$-large sets, and the $k$-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both $\varphi_k(G)$ and $\Omega_k(G)$ and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other things.Comment: 21 page

Едно неравенство за обобщени хроматични графи

Асен Божилов, Недялко Ненов - Нека G е n-върхов граф и редицата от степените на върховете му е d1, d2, . . . , dn, а V(G) е множеството от върховете на G. Степента на върха v бележим с d(v). Най-малкото естествено число r, за което V(G) има r-разлагане V(G) = V1 ∪ V2 ∪ · · · ∪ Vr, Vi ∩ Vj = ∅, , i 6 = j такова, че d(v) ≤ n − |Vi|, ∀v ∈ Vi, i = 1, 2, . . . , r е означено с ϕ(G). В тази работа доказваме неравенството ...Let G be a simple n-vertex graph with degree sequence d1, d2, . . . , dn and vertex set V(G). The degree of v ∈ V(G) is denoted by d(v). The smallest integer r for which V(G) has an r-partition V(G) = V1 ∪ V2 ∪ · · · ∪ Vr, Vi ∩ Vj = ∅, , i 6 = j such that d(v) ≤ n − |Vi|, ∀v ∈ Vi, i = 1, 2, . . . , r is denoted by ϕ(G). In this note we prove the inequality ... *2000 Mathematics Subject Classification: Primary 05C35.This work was supported by the Scientific Research Fund of the St. Kliment Ohridski University of Sofia under contract No 187, 2011

Partitions of graphs into small and large sets

Let GG be a graph on nn vertices. We call a subset AA of the vertex set V(G)V(G)kk-small if, for every vertex v∈Av∈A, deg(v)≤n−|A|+kdeg(v)≤n−|A|+k. A subset B⊆V(G)B⊆V(G) is called kk-large if, for every vertex u∈Bu∈B, deg(u)≥|B|−k−1deg(u)≥|B|−k−1. Moreover, we denote by φk(G)φk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-small sets, and by Ωk(G)Ωk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-large sets. In this paper, we will show tight connections between kk-small sets, respectively kk-large sets, and the kk-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both φk(G)φk(G) and Ωk(G)Ωk(G) and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro–Wei Theorem, Turán’s Theorem and the Hansen–Zheng Theorem among other things.Peer Reviewe