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.