On zero-sum subsequences of length exp(G)

Let $G$ be a finite abelian group. Let $g(G)$ be the smallest positive integer $t$ such that every subset of cardinality $t$ of the group $G$ contains a subset of cardinality $\mathrm{exp}(G)$ whose sum is zero. In this paper, we show that if X is a subset of $\mathbb{Z}^2_{2n}$ with cardinality $4n+1$ and $2n$ or $2n-1$ elements of $X$ have the same first coordintes, then $X$ contains a zero sum subset. As an application of our results we prove that $g(\mathbb{Z}^2_6) = 13.$ This settles Gao-Thangaduri's conjecture for the case $n=6.$ We also prove some results towards the general even $n$ cases of the conjecture.