On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in 2(O)(root n) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no 2(o)(root n)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Exponential Time Hypothesis, for any fixed q, q-Colouring does not admit a 2(o)(root n)-time algorithm, even when restricted to unit disk graphs, and it is solvable in 2O(root n)-time on disk graphs.