Extremal product-one free sequences and $\mathrm{\mid }G\mathrm{\mid }$-product-one free sequences of a metacyclic group

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$thesmallestinteger$t$suchthateverysequenceof$t$elementsin$G$containsaproduct-onesubsequenceoflength$|G|$. In 1961, Erd\H{o}s, Ginzburg and Ziv proved that $\mathsf{E}(G)\le 2\mathrm{\mid }G\mathrm{\mid }-1$ for every finite abelian group $G$ and this result is known as the Erd\H{o}s-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that $\mathsf{E}(G)=\mathsf{d}(G)+\mathrm{\mid }G\mathrm{\mid }$ for every finite group, where $\mathsf{d}(G)$ is the small Davenport constant. Very recently, we confirmed this conjecture for the case when $G=\langle x, y| x^p=y^m=1, x^{-1}yx=y^r\rangle$where$p$isthesmallestprimedivisorof$|G|$and$\mbox{gcd}(p(r-1), m)=1$. In this paper, we study the associated inverse problems on $\mathsf{d}(G)$ and $\mathsf{E}(G)$. Our main results characterize the structure of any product-one free sequence with extremal length $\mathsf{d}(G)$, and that of any $\mathrm{\mid }G\mathrm{\mid }$-product-one free sequence with extremal length $\mathsf{E}(G)-1$.