On Strongly Spanning Trailable Graphs

In this paper,we mainly study the related contents of strongly spanning trail-able graphs.Suppose that e= u1v1 and e'=u2v2 are two edges of G.If e? e',then G(e,e')is the graph obtained from G by replacing e = u1v1 with a path u1vev1 and by replacing e'= u2v2 with a path u2ve'v2,where ve,ve' are two new vertices not in V(G).If e = e',then G(e,e'),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1 A graph G is strongly spanning trailable if for any e,e'? E(G),G(e.e')has a spanning(ve,ve')-trail.By defini-tion,strongly spanning trailable graphs are a special class of supereulerian graphs.Strongly spanning trailable graphs have several useful applications.Shao[Shao Y.Claw-free graphs and line graphs.West Virginia:West Virginia University,2005.]indicated that spanning trailable graphs have applications in the investigation of hamiltonian-connected line graphs.Therefore,carrying on the deep analysis of the research has important practical significance The main conclusions of this paper are as follows:First,we characterize the strongly spanning trailable graphs in graph family C2(4,k).For integers h,l and k with 2 ?h ? 3,l>0 and k>0.let Ch(l,k)denote the family of h-edge-connected simple graphs such that.G ? Ch(l,k)if and only if for every edge cut X C E(G)with size 2 or 3,each component of G-X has at least(|V(G)|-k)/l vertices.We prove that if G ? C2(4,k)and |V(G)| ? 5k+5,then G is not strongly spanning trailable if and only if there exist e,e' ? E(G)such that G(e,e')is contractible to a member in a finite family F.When k=4,F is determined.Second,we summarize research development of degree sequences and strongly spanning trailable graphs.A sequence d =(d1,d2,…,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a re-alization G which is a strongly spanning trailable graph,and d is line hamiltonian-connected if d has a realization G such that.L(G)is hamiltonian-connected.We prove that a nonincreasing multigraphic sequence d =(d1,d2,…,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n?2 and dn?3.Also,we prove that,if a nonincreasing multigraphic sequence d =(d1,d2,…,dn)is strongly spanning trailable except n =1,then d is line hamiltonian-connected.The thesis is divided into four chapters.In Chapter 1,we introduce research background of supereulerian graphs,strongly spanning trailable graphs and some notations of graphs.In Chapter 2,we characterize the strongly spanning trailable graphs in graph family C2(4,k).In Chapter 3,we summarize research develop-ment of degree sequences and strongly spanning trailable graphs.Then we mainly present the proof of main result.Finally,a summary is given and some problems to be further studied are discussed.