  Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$
and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping
$\varphi:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if
it satisfies the condition: for each $v\in V(D)$, there is at least one color
with an odd number of occurrences on each non-empty semi-cut of $v$. We call
the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2
colors, use $def(D)$ to denote the defect of $D$, the minimum number of
vertices in $D$ at which the above condition is not satisfied. In this paper,
we give a descriptive characterization about the weak-odd chromatic index and
the defect of semicomplete digraphs and extended tournaments, which generalize
results of tournaments to broader classes. And we initiated the study of
weak-odd edge covering on digraphs.
