Cubic Cayley graphs of girth at most six and their hamiltonicity
Elham Aboomahigir, Roman Nedela · Acta Mathematica Universitatis Comenianae · 2019
Thomassen's conjecture states that a cubic graph with sufficiently large cyclic connectivity is hamiltonian. Even the following strong conjecture could hold: A cyclically 7-connected cubic graph is hamiltonian, or it is the Coxeter graph. Assuming the conjecture holds true, to prove the hamiltonicity of cubic Cayley graphs it is sufficient to examine cubic Cayley graphs of girth at most 6. Motivated by this, we characterise cubic Cayley graphs of girth at most six and identify few hard families of cubic Cayley graphs of small girth for which we are not able to verify whether they are hamiltonian.