Hamiltonian properties of locally connected graphs with bounded vertex degree

We consider the existence of Hamiltonian cycles for the locally connected graphs with a bounded vertex degree. For a graph GG, let Δ(G)Δ(G) and δ(G)δ(G) denote the maximum and minimum vertex degrees, respectively. We explicitly describe all connected, locally connected graphs with Δ(G)⩽4Δ(G)⩽4. We show that every connected, locally connected graph with Δ(G)=5Δ(G)=5 and δ(G)⩾3δ(G)⩾3 is fully cycle extendable which extends the results of Kikust [P.B. Kikust, The existence of the Hamiltonian circuit in a regular graph of degree 5, Latvian Math. Annual 16 (1975) 33–38] and Hendry [G.R.T. Hendry, A strengthening of Kikust’s theorem, J. Graph Theory 13 (1989) 257–260] on full cycle extendability of the connected, locally connected graphs with the maximum vertex degree bounded by 5. Furthermore, we prove that problem Hamilton Cycle for the locally connected graphs with Δ(G)⩽7Δ(G)⩽7 is NP-complete.