A lower bound for the smallest uniquely hamiltonian planar graph with minimum degree three

Benedikt Klocker, Herbert Fleischner and Günther R. Raidl

Applied Mathematics and Computation, 2020, vol. 380, issue C

Abstract: Bondy and Jackson conjectured in 1998 that every planar uniquely hamiltonian graph must have a vertex of degree two. In this work we verify computationally Bondy and Jackson’s conjecture for graphs with up to 25 vertices. Using a reduction we search for graphs that contain a stable fixed-edge cycle or equivalently a stable cycle with one vertex of degree two. For generating candidate graphs we use plantri and for checking if they contain a stable fixed-edge cycle we propose three approaches. Two of them are based on integer linear programming (ILP) and the other is a cycle enumeration algorithm. To reduce the search space we prove several properties a minimum planar graph with minimum degree at least three containing a stable fixed-edge cycle must satisfy, the most significant being triangle freeness. Comparing the three algorithms shows that the enumeration is more effective on small graphs while for larger graphs the ILP-based approaches perform better. Finally, we use the enumeration approach together with plantri to check that there does not exist a planar graph with minimum degree at least three which contains a stable fixed-edge cycle with 24 or fewer vertices.

Keywords: Uniquely hamiltonian graphs; Integer Linear programming; Stable cycles; Minimum counterexample (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:380:y:2020:i:c:s0096300320302022

DOI: 10.1016/j.amc.2020.125233

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