Saturation numbers of K₃ Pₖ

A graph G is said to be H-saturated if G contains no copy of H, but $G+e$ contains a copy of H for any edge e of Ḡ. The saturation number of H, denoted by sat(n, H), is the minimum number of edges among all H-saturated graphs of order n. In this paper, we investigate the saturation number of K₃ Pₖ for k ≥ 3 and show that sat(n, K₃ Pₖ) = 3n - 6 + sat(n-3, Pₖ) holds for all n ≥ aₖ + 3 and k ≥ 3, which verifies the conjecture [Discrete Appl. Math, 394(2026) 62-80] on sat(n, Kₛ Pₖ) is true for the case $s=3$, where aₖ = k for 3 ≤ k ≤ 5, aₖ = 3 · 2ᵗ⁻¹ - 2 for k = 2t ≥ 6, and aₖ = 2ᵗ⁺¹ - 2 for k = 2t + 1 ≥ 7.