Using Integer Programming to Search for Counterexamples: A Case Study
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It is known that there exist 4-regular, 1-tough graphs which are non-hamiltonian. The smallest such graph known has \(n=18\) nodes and was found by Bauer et al., who conjectured that all 4-regular, 1-tough graphs with \(n\le 17\) are hamiltonian. They in fact proved that this is true for \(n\le 15\), but left open the possibility of non-hamiltonian graphs of 16 or 17 nodes. By using ILP for modeling a counterexample, and then finding out that the model has no solutions, we give an algorithmic proof that their conjecture was indeed correct.
This research has been carried out in the framework of the departmental research project ICON: Innovative Combinatorial Optimization in Networks, Department of Mathematics, Computer Science and Physics (PRID 2017–2018), University of Udine, Italy.
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