Hamiltonian path saturated graphs with small size

A graph G is said to be hamiltonian path saturated (HPS for short), if G has no hamiltonian path but any addition of a new edge in G creates a hamiltonian path in G. It is known that an HPS graph of order n has size at most and, for n?6, the only HPS graph of order n and size is K_n-1∪K₁. Denote by sat(n,HP) the minimum size of an HPS graph of order n. We prove that sat(n,HP)?⌊(3n-1)/2⌋-2. Using some properties of Isaacs’ snarks we give, for every n?54, an HPS graph G_n of order n and size ⌊(3n-1)/2⌋. This proves sat(n,HP)?⌊(3n-1)/2⌋ for n?54. We also consider m-path cover saturated graphs and P_m-saturated graphs with small size.