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The 3-path \(P_3(G)\) of a connected graph \(G\) of order 3 or more has the set of all 3-path (path of order 3) of \(G\) as its vertex of \(P_3(G)\) are adjacent if they have a 2-path in common. A Hamiltonian walk in a nontrivial connected graph \(G\) is a closed walk of minimum length that contains every vertex of \(G\). With the aid of spanning trees and Hamiltonian walks in graphs, we provide sufficient conditions for the 3-path graph of a connected graph to be Hamiltonian.