Hamiltonian Properties And Bipancyclicity Of Bi-Cayley Graphs On Finite Abelian Groups

Let G be a finite group, S(possibly, contains the identity element) be a subset of G. The Bi-Cayley graph BC(G, S) is a bipartite graph with vertex set G × {0,1} and edge set {{(g,0),(sg,1)}, g∈G, s ∈ S}.Let X be a graph. A Hamilton cycle of X is a cycle that contains every vertex of X.Let X be a graph, |V(X)| = n. X is pancyclic if X contains cycles of lengthk(k=3,4,…,n).Let X be a graph, |V(X)| = n. X is bipancyclic if X contains even-cycles of length 2k(k = 2,3,… , ).Let BC(G, S) be a Bi-Cayley graph. An edge of BC(G, S) with vertices (g, 0) and (sg, 1) is called a s edge.A Bi-Cayley graph BC(G, S) is said to be s edge-transitive if for every two s edges e₁ and e₂ of BC(G, S), there is an automorphism of BC(G, S) that maps e₁ to e₂, respectively.In this thesis, we characterize the Hamiltonian properties and bipancyclicity of connected Bi-Cayley graphs on finite Abelian groups. The following are our main results.1. (Lemma 1) Let G be a finite Abelian group. S G, S^-1 = S, S = {S₁,S₂,S₃, … , S_n}, S' = {e, S₂S₁, S₃S₁, … , S_nS₁}, where S₁ is an element of order 2. then (S')^-1 = S' and BC(G,S) ≌ BC{G,S').2. (Lemma 2) Let G be a finite Abelian group. S G, e ∈ S, Bi-Cayley graph BC(G,S) is connected if and only if...