Note on Hamiltonian Graphs in Abelian 2-Groups
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Authors: K. TABAK
DOI: 10.46793/KgJMat2503.401T
Abstract:
We analyze a graph G whose vertices are subgroups of ℤ2k isomorphic to ℤ2 × ℤ2. Two vertices are joined if their respective subgroups have nontrivial intersection. We prove that such a graph is 6(2k−2 − 1)-regular. If a graph is regular, a classical theorem by Ore claims that a graph is Hamiltonian if the degree of any vertex is at least one half of the number of vertices. Using Ore’s theorem, we show that G is Hamiltonian for k ∈{3, 4}. Ore’s theorem cannot be applied when k ≥ 5. Nevertheless, we manage to construct a Hamiltonian cycle for k = 5. Our construction uses orbits of one ℤ24 group under an action of an automorphism of order 31. It is highly likely that this approach could be generalized for k > 5.
Keywords:
Hamiltonian graph, graph, elementary Abelian group, subgroup, group ring.
References:
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