Abstract
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)⩽⌊ 2 3 n⌋ . We prove the sharper result that p(G)⩽ 1 2 u+⌊ 2 3 g⌋ where u is the number of odd vertices and g is the number of nonisolated even vertices.
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