Gallai's conjecture for disconnected graphs

The path number p(G) of a graph G is the minimum number of paths needed to partition the edge set of G. Gallai conjectured that p(G) b n+1 2 c for every connected graph G of order n. Because of the graph consisting of disjoint triangles, the best one could hope for in the disconnected case is p(G) b 2 3 nc. We prove the sharper result that p(G) 1 2 u + b 2 3 gc where u is the number of odd vertices and g is the number of nonisolated even vertices.