Gallai's path decomposition conjecture

Let $G$ be a connected graph on $n$ vertices, and let $p(G)$ denote the minimum number of paths in a path decomposition of $G$. Gallai's conjecture.

$$p(G)\leq \frac{n+1}{2}.$$

This conjecture is a central problem on path decompositions; the paper notes that it is difficult for graphs with many vertices of even degree and discusses known results for general graphs and Eulerian graphs of maximum degree at most $4$.