Cycles In Locally Hamiltonian And Locally Hamilton-connected Graphs

Let P be a property of a graph.A graph G is said to be locally P,if the subgraph induced by the open neighbourhood of every vertex in G has property P.Ryjacek conjectures that every connected,locally connected graph is weakly pancyclic.Motivated by the above conjecture,van Aardt et al.[S.A.van Aardt,M.Frick,O.R.Oellermann,J.P.de Wet,Global cycle properties in locally connected,locally traceable and locally hamiltonian graphs,Discrete Appl.Math.205(2016)171-179]investigated the global cycle structures in connected,locally connected/trace-able/hamiltonian graphs.Among other results,they proved that a connected,locally hamiltonian graph G with maximum degree at least |V(G)|-5 is weakly pancyclic.In the second chapter,we improve this result by showing that such a graph with maximum degree at least |V(G)|-6 is weakly pancyclic.Furthermore,we show that a connected,locally Hamilton-connected graph with maximum degree at most,7 is fully cycle extendable,in[10],Wet and van Aardt prove that if a connected,locally hamiltonian graph G with n? 13 is traceable.In the third chapter,by restricting the number of vertex and maximum degree,we obtain that a connected,locally hamiltonian graph with order n? 10 and ?(G)? 7 is fully cycle extendable.Finally,as a consequence of the above results,we obtain that a connected,locally hamiltonian graph with order n ? 10 is pancyclic.