Gallaiʹs conjecture for disconnected graphs Original Research Article

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)⩽⌊(n+1)/2⌋ for every connected graph G of order n. Because the graph consisted of disjoint triangles, the best one could hope for in the disconnected case is p(G)⩽⌊23n⌋. We prove the sharper result that p(G)⩽12u+⌊23g⌋ where u is the number of odd vertices and g is the number of nonisolated even vertices.