Research About Hamiltonian Property Of Special Graph

Algebraic graph theory is an important branch of the research on discrete mathematics,and it is mainly to solve the problem of graph theory by means of algebraic methods(such as group theory,etc).The study of the Hamiltonian property of graph is a hot topic in this branch.But it has not yet been completely solved.In this paper,we mainly investigate the Hamiltonian property of special graph by use of the theory of abstract group and some methods or skills of graph theory.First of all,we focus on the study of the Hamiltonian property of bi-Cayley graph.The problem which is about the existence of Hamilton cycle in a bi-Cayley graph on special group G is transformed into the other one which is about the structure of finite group.The realization of this process is through the introduction of strong S-staggered sequence corresponding to the path and cycle of bi-Cayley graph.For a bi-Cayley graph,its Hamiltonian cycles are constructed by those Hamiltonian cycles of the corresponding(single)Cayley graph,of the bi-Cayley graph on G’s quotient group,and of its induced subgraphs.Furthermore,some results about the Hamiltonian property of bi-Cayley graphs on the group which has order pq(p>q>2 are prime)or is the generalized quaternion group Q4r(r is odd prime)are obtained.Secondly,with the help of GAP,we study the Hamiltonian property of a connected Cayley digraphs Cay(GH,S)on a class of finite non-abelian group Gh =<H,σ|σ2n = 1,σ-1hσ = h-1,(?)h∈H)with cyclic commutator subgroup H.Two sufficient conditions of such connected Cayley digraphs are obtained.As ap-plications,we also give two examples of constructing Hamilton cycle in Cayley digraphs on the group which is the generalized bicyclic group and is the general-ized dihedral group via recursive algorithm.