Eulerian odd-cover conjecture

Let $G$ be an Eulerian graph, let $\Delta(G)$ be its maximum degree, and let a path odd-cover (respectively, cycle odd-cover) be a collection of paths (respectively, cycles) whose symmetric difference is $G$. Eulerian odd-cover conjecture. There exists a constant $c$ such that $G$ admits a path odd-cover of size

$$\frac{1}{2}\Delta(G)+c$$

and a cycle odd-cover of size at most

$$\frac{1}{2}\Delta(G)+c.$$

The paper proves upper bounds of $\lceil3\Delta(G)/4\rceil$ for both types of odd-cover, whereas the trivial lower bound is approximately $\Delta(G)/2$; the conjecture proposes that only an additive constant separates the optimum from this lower bound.