Yuansheng Yang, Jianhua Lin, Chunli Wang,and Kaifeng Li. On Kotzig's conjecture concerning graphs with a unique regular path-connectivity. Perhaps the problem is still open. Nathaniel Dean and Mekkia Kouider. Gallai's conjecture for disconnected graphs.
G is the minimum number of paths needed to partition the edge set of G .
G on n vertices is the union of n-1 cycles. Zoltan Furedi, On the number of edges of quadrilateral-free graphs. There, reference to an earlier work of Pyber is also given. This was obtained with the aid of a computer program. Julio C. Subocz G., Some Values of Olson's Constant .