Some Results Concerning Hamiltonicity Of Graphs

It is well-known that the problem on hamiltonicity of graphs is a very important problem in graph theory and the research is very active. There are a number of papers dealing with this problem each year. In this thesis, we studied the hamiltonicity of graphs systematically and proved several new theorems which are the improvements and generalizations of some known results on this topic. And some of these results are the best possible in some sense .The main new results of this thesis are the following.(1). Let s be a positive integer and G a s -connected graph of order n. If d(S) + d(T) > n for every two strongly disjoint independent sets S and T with S = s and T = 1, respectively, then G is hamiltonian.(2). Let s and t be two positive integers and G a 2(s + t) -connected graph of order n. If d(S) + d(T) > n for every two strongly disjoint independent sets S and T with S = s and T = t, respectively, then G is hamiltonian.(3). Let s and t be two positive integers and G a (s + t)-connected graph of order n ?s + t (n > (s + t)3 - (s + t}). If d(S) + d(T) > n for every two strongly disjoint independent sets S and T with S ?s and T = t, respectively, then G is hamiltonian.(4). Let G be a k -connected graph with k > 2, b an integer, and 0 < 6 < k. Ifin G for each then G is hamiltonian.(5). Let G be a k-connected graph with k >2, b an integer, and 0 < b < k + 1.Ifin G for each then G is hamiltonian.(6). Let G be a k-connected graph with k >2, b an integer, and 0 < b < k + l. Set u = 1 (when 1 < 6 < k), or u = 0 (when b e {1, k}). Ifi=0in G for each then G is hamiltonian.(7). Let G be a k -connected graph with k > 2, b an integer, I < b < k, and n(Z) large enough (i.e.,n(Z) > b(2k - b + 1) - 1). //in G for each then G is hamiltonian.(8). Let G be a k-connected graph with k > 2, b an integer, and 1 < b < k. If then G is hamiltonian.(9). Let s, t, w be three positive integers and Ga(s + t + w -1l)-connected claw-free graph of order n. Iffor every three strongly disjoint independent sets S, T and W with S = s, T = t, and W = w, respectively, then G is hamiltonian.In addition, the corresponding results on traceable, hamilton-connected or 1-hamiltonian are obtained, too.