Naslov
NOTE ON HAMILTONIAN GRAPHS IN ABELIAN 2-GROUPS

Autori
Kristijan Tabak

Izvornik
Kragujevac journal of mathematics (1450-9628) 49 (2022), 3; 401-409

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Hamiltonian graph ; graph ; elementary Abelian group ; subgroup ; group ring

Sažetak
We analyze a graph $G$ whose vertices are subgroups of $\mathbb{;Z};_2^k$ isomorphic to $\mathbb{;Z};_2 \times \mathbb{;Z};_2.$ Two vertices are joined if their respective subgroups have nontrivial intersection. We prove that such a graph is $6(2^{;k-2};-1)$-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 $k \in \{;3, 4\};.$ Ore's theorem cannot be applied when $k \geq 5$ . Nevertheless, we manage to construct a Hamiltonian cycle for $k=5.$ Our construction uses orbits of one $\mathbb{;Z};_2^4$ group under an action of an automorphism of order $31.$ It is highly likely that this approach could be generalized for $k>5.$

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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