Zero-sum partition conjecture for finite Abelian groups with multiple involutions

From papers

Let Γ\Gamma be a finite Abelian group of order nn with more than one involution. Let

n1=r1+r2++rtn-1=r_1+r_2+\cdots+r_t

be a partition of n1n-1 with ri3r_i\geq 3 for 1it1\leq i\leq t, where tt is any positive integer. Zero-sum partition conjecture. There is a partition of Γ{0}\Gamma\setminus\{0\} into pairwise disjoint subsets A1,A2,,AtA_1,A_2,\ldots,A_t such that

Ai=riandaAia=0|A_i|=r_i\quad\text{and}\quad \sum_{a\in A_i}a=0

for 1it1\leq i\leq t. The conjecture is true for groups isomorphic to (Z2)m(\mathbb{Z}_2)^m, as proved by Egawa; the parser marks this candidate as resolved, although the supplied context does not state whether the full assertion or only that special case has been resolved.

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Sources & referencesView supporting material

Primary source

Sylwia Cichacz and Zsolt Tuza, “Realization of digraphs in Abelian groups and its consequences”, arXiv:1901.08629 (2019).

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