N-Factor-Criticality And Hamilton-Connectivity In Graph Theorey

This thesis mainly concentrates on n - factor - criticality and n - extendablity in graph theory,and we will find the first two chapters that contain more or less independent topics within this research field.In the first chapter we prove the following result:Let G be a graph of order p with p=n(mod 2) and np + n - 1 holds for every independent set S of s vertices in G.In the last chapter,we focus on hamilton - connectivity for every hamilton - connected graph with even order is 0 - factor - critical. Before the discussion of this chapter,we prove the following useful lemma:Let G be a graph of order n,then the inequality d(u) + d(v) + d(w) - N(u) N(v)N(w)>3NC-n+3 holds for any independent set (u,v,w)in V(G).Combination of this lemma and the hamilton - connectivity involving neighborhood intersection obtains and improves or generalizes a series of classical results involving hamilton - connectivity,at the same time,we construct some extremal graphs to show the improved results to be the best possible ones. As for hamiltonicity,we omit the similar discussion.