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 (n-1 2) and, for n ≥ 6, the only HPS graph of order n and size (n-1 2) is Kn-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 Gn 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 Pm-saturated graphs with small size.

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