Local properties of graphs that induce global cycle properties

weakly pancyclicity T57-57.97 Applied mathematics. Quantitative methods fully cycle extendability locally connected
DOI: 10.7494/opmath.2025.45.2.275 Publication Date: 2025-03-10T20:27:12Z
ABSTRACT
A graph \(G\) is locally Hamiltonian if \(G[N(v)]\) is Hamiltonian for every vertex \(v\in V(G)\). In this note, we prove that every locally Hamiltonian graph with maximum degree at least \(|V(G)| - 7\) is weakly pancyclic. Moreover, we show that any connected graph \(G\) with \(\Delta(G)\leq 7\) and \(\delta(G[N(v)])\geq 3\) for every \(v\in V (G)\), is fully cycle extendable. These findings improve some known results by Tang and Vumar.
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