Two New Classes of Hamiltonian Graphs

We prove that a triangular grid without local cuts is (almost) always Hamiltonian. This suggests an efficient scheme for rendering triangulated manifolds by graphics hardware. We also show that the Hamiltonian Cycle problem is NP-Complete for planar subcubic graphs of arbitrarily high girth. As a by-product we prove that there exist tri-Hamiltonian planar subcubic graphs of arbitrarily high girth. 1 Definitions and Notation We say that a graph G = (V,E) is induced by a set S ⊂ R if V = S, and E = {{i, j} | i, j ∈ S, |i− j| = 1}. Let Z∆ be the infinite triangular lattice, i.e., the set of vertices of the tiling of R with unit equilateral triangles. A triangular grid graph, or triangular grid, is a plane graph induced by a subset of vertices of Z∆. Let G = (V,E) be a triangular grid without degree-1 vertices. A bounded face f of G is called a hole if f is not a unit equilateral triangle. Let h denote the number of holes in G. A vertex v ∈ V is called a cut if its removal disconnects G; v is a local cut v is a cut or if the number of holes in G \ v is less than h. A vertex v ∈ V is called boundary if its degree is less than 6. The non-boundary vertices of G (degree-6 vertices) are called internal vertices and denoted by V6. The subgraph B of G induced by its boundary vertices, V \ V6, is a set of cycles B = {C, C1, . . . , Ch}. The cycle C ∈ B that separates G from its unbounded face is called the outer boundary. Each cycle in B \ C = {C1 . . . Ch} bounds a hole, i.e., is the boundary of a hole of G. Having no local cut means that the cycles in B are simple and vertex-disjoint. Let the cost of B be the number of edges in it. For g ∈ N we denote by Gg the class of planar bipartite maximum-degree-3 graphs of girth g. Our Contributions We prove that triangular grids without local cuts are always Hamiltonian, with the exception of one special graph, “The Star of David” (Fig. 1). Our proof is constructive and allows one to find the Hamiltonian cycle in linear time. This has application in computer graphics as it gives an efficient scheme for outputting triangulation data. We prove that for arbitrarily high g, the Hamiltonian cycle problem is NP-complete for graphs from Gg . We also prove that for arbitrarily high g there exist graphs in Gg that have exactly 3 Hamiltonian cycles. 2 Triangular Grids without Local Cuts are Hamiltonian The crucial observations that we use are as follows: 1) One can attach to the cycles in B all internal vertices at the cost of 1 per ∗Applied Math and Statistics, Stony Brook University. †Helsinki Institute for Information Technology. Figure 1: The only non-Hamiltonian triangular grid without local cuts: the Star of David. Cj Ci