  Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A
$T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident
with an odd number of edges in $J$ if and only if the vertex belongs to $T$.
Minimum $T$-joins have many applications in combinatorial optimizations. In
this paper, we show that a minimum $T$-join of a connected graph $G$ has at
most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is
the maximum bidegeless subgraph of $G$. Further, we are able to use this result
to show that every flow-admissible signed graph $(G,\sigma)$ has a
signed-circuit cover with length at most $\frac{19} 6 |E(G)|$. Particularly, a
2-edge-connected signed graph $(G,\sigma)$ with even negativeness has a
signed-circuit cover with length at most $\frac 8 3 |E(G)|$.
