Inverse results for weighted Harborth constants

Luz Elimar Marchan (1) , Oscar Ordaz (2) , Dennys Ramos (1) , Wolfgang Schmid (3)

1 Departamento de Matemáticas, Decanato de Ciencias y Tecnologías, Universidad Centroccidental Lisandro Alvarado
2 Laboratorio MoST, Centro ISYS
3 LAGA - Laboratoire Analyse, Géométrie et Applications

For a finite abelian group $(G,+)$ the Harborth constant is defined as the smallest integer $\ell$ such that each squarefree sequence over $G$ of length $\ell$ has a subsequence of length equal to the exponent of $G$ whose terms sum to $0$. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling $0$ is required, that is, when forming the sum one can chose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constant for certain groups, in particular for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erd\H{o}s--Ginzburg--Ziv constant.

Mots clés: finite abelian group; weighted subsum; zero-sum problem

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Luz Elimar Marchan, Oscar Ordaz, Dennys Ramos, Wolfgang Schmid. Inverse results for weighted Harborth constants. International Journal of Number Theory, 2016, 12 (7), pp.1845-1861. ⟨10.1142/S1793042116501141⟩. ⟨hal-01149390v2⟩