  We introduce and study "path odd-covers", a weakening of Gallai's path
decomposition problem and a strengthening of the linear arboricity problem. The
"path odd-cover number" $p_2(G)$ of a graph $G$ is the minimum cardinality of a
collection of paths whose vertex sets are contained in $V(G)$ and whose
symmetric difference of edge sets is $E(G)$.
  We prove an upper bound on $p_2(G)$ in terms of the maximum degree $\Delta$
and the number of odd-degree vertices $v_{\text{odd}}$ of the form
$\max\left\{{v_{\text{odd}}}/{2}, 2\left\lceil {\Delta}/{2}\right
\rceil\right\}$. This bound is only a factor of $2$ from a rather immediate
lower bound of the form $\max \left\{ {v_{\text{odd}} }/{2} , \left\lceil
{\Delta}/{2}\right\rceil \right\}$. We also investigate some natural
relaxations of the problem which highlight the connection between the path
odd-cover number and other well-known graph parameters. For example, when
allowing for subdivisions of $G$, the previously mentioned lower bound is
always tight except in some trivial cases. Further, a relaxation that allows
for the addition of isolated vertices to $G$ leads to a match with the linear
arboricity when $G$ is Eulerian. Finally, we transfer our observations to
establish analogous results for cycle odd-covers.
