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Local properties of graphs that induce global cycle properties

Wang Yanyan, Yang Xiaojing

Opuscula Mathematica

2025

Vol. 45, no. 2

275--285

A graph G is locally Hamiltonian if G[N(v)] is Hamiltonian for every vertex v ∈ 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 Δ(G) ≤ 7 and δ(G[N(v)]) ≥ 3 for every v ∈ V (G), is fully cycle extendable. These findings improve some known results by Tang and Vumar.