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 (formula presented) 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 (formula presented)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Expo-nential Time Hypothesis, for any fixed q, q-Colouring does not admit a (formula presented)-time algorithm, even when restricted to unit disk graphs, and it is solvable in (formula presented)-time on disk graphs.
CITATION STYLE
Kisfaludi-Bak, S., & Van Der Zanden, T. C. (2017). On the exact complexity of hamiltonian cycle and q-colouring in disk graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10236 LNCS, pp. 369–380). Springer Verlag. https://doi.org/10.1007/978-3-319-57586-5_31
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