On some weighted zero-sum constants

Abstract

Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.

Weighted EGZ-constant for p-groups of rank 2

“…(Induction on β). The cases β = 1 or 2 were proved by Chintamani and Paul in [5,6]. We may then assume β ≥ 3.…”

“…As before we are going to proceed by a simultaneous induction over α and β. Chintamani and Paul proved in [5,6] that this result is true whenever α ≥ β and β ∈ {1, 2}. Hence we will assume that min(α, β) ≥ 3 and also assume that any sequence over Z p γ ⊕Z p δ , with δ+γ < α+β, and of length n = s A (Z p γ ⊕Z p δ )−1, with no A-weighted zero-sum subsequence of length p max(δ,γ) has all the characteristics described above, that is, (5.2)…”

“…Recently, Chintamani and Paul (see [5,6]) proved that s A (Z p s ⊕Z p α ) = p α +α+s and η A (Z p s ⊕ Z p α ) = α + s + 1, for s ∈ {1, 2}, α ≥ s and p an odd prime number. They also classify the extremal A-weighted zero-sum free sequences and extended these results proving that s A (Z p s ⊕ Z n ) ≤ n + Ω(n) + 2s, for s ∈ {1, 2}, provided p s |n.…”

“…We list here some of these contributions. (i) s A (Z r 2 ) = 2 r + 1 (see [11]); (ii) s A (Z 3 ) = 4 (see [3]), s A (Z 2 3 ) = 5 (see [4]), s A (Z 3 3 ) = 9 (see [9,13]), s A (Z 4 3 ) = 21, s A (Z 5 3 ) = 41 and s A (Z 6 3 ) = 113 (see [13]); (iii) s A (Z 4 ) = 6 and s A (Z 6 ) = 8 (see [3]); s A (Z 2 4 ) = 8 (see [1]); (iv) s A (Z 2 ⊕Z 4 ) = 7 (see [14,15]), s A (Z 2 2 ⊕Z 4 ) = 8 (see [15]) and s A (Z 2 ⊕Z 6 ) = 9 (see [14]); The following results relate these two constants.…”

“…(ii) s A (Z 3 ) = η A (Z 3 ) + 3 − 1 (see [3]), s A (Z 2 3 ) = η A (Z 2 3 ) + 3 − 1 (see [4,15]); (iii) s A (Z 4 ) = η A (Z 4 ) + 4 − 1 and s A (Z 6 ) = η A (Z 6 ) + 6 − 1 (see [3]); (iv) s A (Z 2 4 ) = η A (Z 2 4 ) + 4 − 1 (see [1,15]); (v) s A (Z 2 ⊕ Z 4 ) = η A (Z 2 ⊕ Z 4 ) + 4 − 1 (see [14,15]), s A (Z 2 2 ⊕ Z 4 ) = η A (Z 2 2 ⊕ Z 4 ) + 4 − 1 (see [15]) and s A (Z 2 ⊕ Z 6 ) = η A (Z 2 ⊕ Z 6 ) + 6 − 1 (see [14]); (vi) s A (Z p s ⊕ Z p r ) = η A (Z p s ⊕ Z p r ) + p r − 1, where p is an odd prime number and s ∈ {1, 2} (see [5,6]);…”