Path odd-covers of graphs

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 $Δ$ and the number of odd-degree vertices $v_{\text{odd}}$ of the form $\max\left\{v_{\text{odd}}/{2}, 2\left\lceil Δ/{2}\right \rceil\right\}$. This bound is only a factor of $2$ from a rather immediate lower bound of the form $\max \left\{ {v_{\text{odd}} }/{2} , \left\lceil Δ/{2}\right\rceil \right\}$. 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.