Some exact values of the Harborth constant and its plus-minus weighted analogue

Abstract

minimal correctionInternational audienceThe Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order $2$. Moreover, we contrast these results with existing results and conjectures on these problems

Harborth constants for certain classes of metacyclic groups

“…We leave it to the reader to verify that b = 0, b ∈ {0, n 2t }, b ∈ {0, n 2(t+1) } must be added to the equality conditions in the first, second, and third cases, respectively. The following lemma generalizes a well-known result that appears in [3] and [13], among other places. Lemma 2.3.…”

“…So, in a sense, the commutator relation yx = x −1 y builds in enough "flexibility" that the Harborth constant is stable under the introduction of plus-minus weightings. Further analogies to the discussion in [13] could also be of future interest.…”

“…Much less is known about the Harborth constant than about the EGZ constant despite their similarities. Marchan, Ordaz, Ramos, and Schmid [13], [14] computed the Harborth constants for some classes of abelian groups, in particular, g(C 2 ⊕ C 2n ) = 2n + 3, n odd 2n + 2, n even for all n ≥ 1. They also consider the plus-minus weighted analogue of the Harborth constant, in which one may add either an element or its inverse in the sums of length exp(G).…”

“…Connection to the plus-minus weighted analogue. The ability of the powers of x to contribute either positively or negatively to products is reminiscent of the plus-minus weighting discussed in [13]. It is easy to see that g ± (H n,m ) = n + 2 when n ≥ 6 and m are both even.…”

Inverse results for weighted Harborth constants

“…In an earlier work [13] we determined the exact value of g(C 2 ⊕ C 2n ) and of g ± (C 2 ⊕ C 2n ) (we denote by C n a cyclic group of order n). In particular, it turned out that g(C 2 ⊕ C 2n ) = g ± (C 2 ⊕ C 2n ) = 2n + 2 for even n ≥ 4; we recall the complete result in Section 5.…”