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

Given a graph G = (V, E), V(G) = V and E(G) = E denote the vertex set and the edge set of G, respectively. All graphs considered in this paper are undirected graphs. A simple path (or path for short) is a sequence of adjacent edges (v1, v2), (v2, v3), ..., (vm-2, vm-1), (vm-1, vm), written as 〈v1, v2, v3, ..., vm〉, in which all of the vertices v1, v2, v3, ..., vm are distinct except possibly v1 = vm. The path 〈v1, v2, v3, ..., vm〉 is also called a cycle if v1 = vm and m ≥ 3. A cycle that traverses every vertex in the graph exactly once is called a hamiltonian cycle. A graph that contains a hamiltonian cycle is called a hamiltonian graph or said to be hamiltonian. A graph is edge hamiltonian if each edge in the graph is incident with some hamiltonian cycle in the graph. The diameter of graph G is the maximum distance among all pairs of vertices in G, where distance means the length of a shortest path joining two distinct vertices in G. For V' ⊂ V and E' ⊂ E, G − V' − E' denotes the graph obtained by removing all of the vertices in V' from V and removing the edges incident with at least one vertex in V' and also all of the edges in E' from E. Let k be a positive integer. A graph G is k-hamiltonian if G − V' − E' is hamiltonian for any set V' ⊂ V and E' ⊂ E with |V'| + |E'| ≤