Existence conditions for barycentric sequences

Abstract

Let G be a ÿnite abelian group. A sequence in G is barycentric if it contains one element which is the "average" of its terms. We give some su cient conditions for the existence of barycentric sequences, with prescribed or unconstrained length.

Some remarks on barycentric-sum problems over cyclic groups

“…The latter part, establishing the upper bound k + 2 is essentially [5,Theorem 8]; it is possible to replace this part by invoking the more general result Theorem 4.4, obtained by quite different means, yet we avoid to do this to illustrate the merits of the different methods. Indeed, we did obtain an argument along the lines of [5,Theorem 8] for establishing the upper bound of k + 2 for k = (p − 3)/2 as well, yet the details become quite tedious so that we feel that to push this method too far further is infeasible, which necessitates an other approach, whence we subsequently establish Theorem 4.4. In the present case only parts of the argument would, under similar technical conditions as in Theorem 4.2, carry over to cyclic groups of composite order, so that we do not make this explicit.…”

“…It remains to show that BO(k, Z/pZ) ≤ k + 2; the argument is essentially [5,Theorem 8]. Let C ⊂ Z/pZ a set with k + 2 elements.…”

Un método algoritmo para el cálculo del número baricéntrico de Ramsey para el grafo estrella

“…El estudio de las secuencias k-baricéntricas se inician en [5] y [6]. En [6] la constante de Davenport Baricéntrica, BD(G), es decir, el menor entero positivo t tal que toda t-secuencia en G contiene una subsecuencia baricéntrica.…”

“…El estudio de las secuencias k-baricéntricas se inician en [5] y [6]. En [6] la constante de Davenport Baricéntrica, BD(G), es decir, el menor entero positivo t tal que toda t-secuencia en G contiene una subsecuencia baricéntrica. En [5] la constante de Davenport k-baricéntrica, BD(k, G), se define como el menor entero positivo t tal que toda t-secuencia en G contiene una subsecuencia k-baricéntrica.…”

On the Barycentric Labeling of Certain Graphs

“…Note that the motivation of Definition 4 is the following definition of -barycentric sequence which was introduced in [9] and has already been used in graph labeling problems, specially in Ramsey theory [9][10][11]. Example 6.…”