Hamiltonian Properties Of 4-Connected Claw-free Graphs And Some Related Problems

In this dissertation we study the Hamiltonian sufficient conditions for K1,3-free graphs and give a new sufficient conditions for Hamiltonian cycles in such graphs.It shows that if d(P5)+d(P1)?n-2 holds for every two disjoined P5 and P1 in a 4-connected K1,3-free graph G of order n,then G has a Hamiltonian cycle.Further,we discuss the structure and properties of 4-cycles in bipartite graphs and obtain the extremal graphs by using a new counting formula.