A Diametric Theorem for Edges

A diametric theorem for edges

The Edge-Diametric Theorem in Hamming Spaces

The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A 92(1) (2000) 1–16]. © 2007 Elsevier B.V. All rights reserved.

The Diametric Theorem in Hamming Spaces-Optimal Anticodes

Ž n . For a Hamming space X , d , the set of n-length words over the alphabet a H 4 n X s 0, 1, . . . , a y 1 endowed with the distance d , which for two words x s a H Ž . n Ž . n x , . . . , x , y s y , . . . , y g X counts the number of different components, 1 n 1 n a we determine the maximal cardinality of subsets with a prescribed diameter d or, in another language, anticodes with distance ...

A Diametric Theorem in Z nm for Lee and Related Distances

We present the diametric theorem for additive anticodes with respect to the Lee distance in Zn2k , where Z2k is an additive cyclic group of order 2. We also investigate optimal anticodes in Znpk for the homogeneous distance and in Zm for the Krotov-type distance.

A stability theorem on fractional covering of triangles by edges

Let ν(G) denote the maximum number of edge-disjoint triangles in a graph G and τ(G) denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that τ(G) ≤ 2ν(G) for every graph G. This is sharp, since for the complete graphK4 we have ν(K4) = 1 and τ (K4) = 2. We refine this result by showing that if a graph G has τ(G) ≥ 2ν(G) − x, then G contains ν(G)...