  We study the maximal cross number $\mathsf{K}(G)$ of a minimal zero-sum
sequence and the maximal cross number $\mathsf{k}(G)$ of a zero-sum free
sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the
first part of this paper, we extend a previous result by X. He to prove that
the value of $\mathsf{k}(G)$ conjectured by Krause and Zahlten hold for $G
\bigoplus C_{p^a} \bigoplus C_{p^b}$ when it holds for $G$, provided that $p$
and the exponent of $G$ are related in a specific sense. In the second part, we
describe a new method for proving that the conjectured value of $\mathsf{K}(G)$
hold for abelian groups of the form $H_p \bigoplus C_{q^m}$ (where $H_p$ is any
finite abelian $p$-group) and $C_p \bigoplus C_q \bigoplus C_r$ for any
distinct primes $p,q,r$. We also give a structural result on the minimal
zero-sum sequences that achieve this value.
