On joint short minimal zero-sum subsequences over finite abelian groups of rank two

Let $(G,+,0)$ be a finite abelian group and let ${\eta }^N\left(G\right)$ be the smallest integer ℓ such that every sequence over $G\setminus \left\{0\right\}$ of length ℓ has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that ${\eta }^N(C_n\oplus C_n)=3n+1$ for every $n\ge 2$ and solved the corresponding inverse problem for groups $C_p\oplus C_p$, where p is a prime. In this paper, we determine the precise value of ${\eta }^N\left(G\right)$ for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups $C_n\oplus C_n$, where $n\ge 2$, which confirms a conjecture of Gao, Geroldinger and Wang for all $n\ge 2$ except $n=4$.