Abstract
Dirac’s theorem (1952) is a classical result of graph theory, stating that an n-vertex graph (\(n \ge 3\)) is Hamiltonian if every vertex has degree at least n/2. Both the value n/2 and the requirement for every vertex to have high degree are necessary for the theorem to hold.
In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian Cycle problem can be solved in time \(c^k \cdot n^{O(1)}\), for a fixed constant c, if at least \(n-k\) vertices have degree at least n/2, or if all vertices have degree at least \(n/2 - k\). The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH).
The results extend the range of tractability of the Hamiltonian Cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with O(k) vertices can be found in polynomial time.
A full version of the paper is available on arXiv [26].
B.M.P. Jansen—Supported by NWO Gravitation grant “Networks”.
L. Kozma—Supported by ERC Consolidator Grant No 617951.
J. Nederlof—Supported by NWO Gravitation grant “Networks” and NWO Grant No 639.021.438.
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Acknowledgement
We thank Naomi Nishimura, Ian Goulden, and Wendy Rush for obtaining a copy of Bondy’s 1980 research report [8].
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Jansen, B.M.P., Kozma, L., Nederlof, J. (2019). Hamiltonicity Below Dirac’s Condition. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_3
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