We give a systematic study of Hamiltonicity of grids-the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, "square", grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomial-time algorithm we give efficient algorithms to find a Hamiltonian cycle.We also establish, for any g ≥ 6, a one-to-one correspondence between Hamiltonian cycles in planar bipartite maximum-degree-3 graphs and Hamiltonian cycles in the class C g of girthg planar maximum-degree-3 graphs. As applications of the correspondence, we show that for graphs in C g the Hamiltonian cycle problem is NP-complete and that for any N ≥ 5 there exist graphs in C g that have exactly N Hamiltonian cycles. We also prove that for the graphs in C g , a Chinese Postman tour gives a (1 + 8 g )-approximation to TSP, improving thereby the Christofides ratio when g > 16. We show further that, on any graph, the tour obtained by Christofides' algorithm is not longer than a Chinese Postman tour.

A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5,4,6,14]. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple k-factors for all values of k ≥ 2.

We examine a recolouring scheme ostensibly used to assist in classifying geographic data. Given a drawing of a graph with bi-chromatic points, where the points are the vertices of the graph, a point can be recoloured if it is surrounded by neighbours of the opposite colour. The notion of surrounded is defined as a contiguous subset of neighbours that span an angle greater than 180 degrees. The recolouring of surrounded points continues in sequence, in no particular order, until no point remains surrounded. We show that for some classes of graphs the process terminates in a polynomial number of steps. On the other hand, there are classes of graphs with infinite recolouring sequences.