On Hamilton Cycles in Locally Connected Graphs with Vertex Degree Constraints

It is shown that every connected, locally connected graph with the maximum vertex degree Δ(G)=5 and the minimum vertex degree δ(G)⩾3 is fully cycle extendable. For Δ(G)⩽4, all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ(G)⩽7 is shown to be NP-complete.