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A Family of Trivalent 1-Hamiltonia Graphs With Diameter O(log n)

In this paper, we construct a family of graphs denoted by Eye(s) that are 3-regular, 3-connected, planar, hamiltonian, edge hamiltonian, and also minimal 1-hamiltonian. Furthermore, the diameter of Eye(s )i sO(log n), where n is the number of vertices in the graph and to be precise, n =6 (2 s − 1) vertices.