1 ON THE HAMILTONICITY OF MAXIMAL PLANAR GRAPHS I. CAHIT icahit@mail.ebim.com.tr Department of Computer Science and Engineering European University of Lefke, Lefke, North Cyprus M. OZEL mehmetozel@emu.edu.tr Department of Mathematics Eastern Mediterranean University ABSTRACT. By the introduction of fundemental graphs, the sources of the non- Hamiltonicity of maximal planar graphs have been identified, and a necessary and a sufficient condition has been given for maximal planar graphs to be non-Hamiltonian. Keywords: Hamiltonicity, Maximal Planar graphs, Outerplanar graphs 1. Introduction Although there has been a large amount of work done on the hamiltonicity of graphs, few results have been reported specifically on the hamiltonicity of maximal planar graphs. Maximal planar graphs have the property that the addition of any other edge results with a nonplanar graph and the planar drawing of a maximal planar graph is such that the boundaries of every one of its faces are a cycle of length three [1]. A graph is Hamiltonian if it contains a cycle that passes through every vertex of the graph exactly once. If asked about the existence of a non-hamiltonian maximal planar graph, the most frequent response would be, “in order to find one insert a vertex into each face of a octahedron and connect it to the surrounding vertices.” This process would produce a non-hamiltonian maximal planar graph but why? Would only several insertions be sufficient or which insertions would really be necessary? By the introduction of fundemental graphs (fgs) in the first part of the paper, a groundwork has been prepared for the addressing of these issues. In the following section, a necessary and a sufficient condition has been introduced for a maximal planar graph to be non-hamiltonian. The proofs provide a significant insight regarding the reasons for the non-hamiltonicity of the maximal planar graphs and regarding the edge where a maximal planar graph shifts from hamiltonicity to non-hamiltonicity or vice versa. Finding the Hamiltonian cycle in certain graphs, such as planar bipartite graphs, perfect graphs and grid graphs has been shown to be NP complete. Although, Hamiltonian cycle problem is NP-complete for 3-connected planar graphs, Chiba and Nishizeki has presented a polynomial time algorithm for 4-connected planar graphs [2]. It has been proved by Garey, Johnson and Tarjan that the Hamiltonian cycle problem is NP-