This paper discusses the application of degree in other aspects of graph theory,such as cycle,Hamiltonian graph and matching on the base of reference.

In this paper,we will research the domination number of hamiltonian graphs and prove that for a hamiltonian graph G of order n with minimun degree at least five,the domination munber of is at most 5n/14.

Then Tournament D *=(V,E) is Hamiltonian graph.

One Sufficient Condition of Making D—Circle Graph Be a Hamilton Graph;

This paper mainly concerns the properties of Hamilton graph and some methods of judgment based on them.

In 1998 a conjecture was suggested for the conference of Graph theory,combinatorics,and applications at Kalamazooin USA as follows:let G be a 3-connected K1,3-free graph of order n,if |N(x)∪N(y)|≥(2n-6)/3 for each pair of nonadjacent vertices x,y,then G is Hamiltonian.

If G is Hamiltonian,then its line graph L(G) is pancyclic graph.

These results imply several known theorems on the topic of hamiltonian graph, hamilton connected graph.

By using Hamilton graph,the problem of wagons placing-in and taking-out on branch-shaped sidings can be turned into searching Hamilton loop of minimum power.

he concept of the degree of a face is introduced, associated theoreme as necessary conditions of Hamilton graph are presented, and a method to search for Hamilton circuits in a given connected planer graphabsorbing transformation of graph-is Put forward.

By adopting Hamilton Graph, the paper builds up a mathematical model for wagons’ placing-in and taking-out and goods loading & unloading operation.

If a connected graph G is 1-hamiltonian(contains a 2factor with k(k2) cycles、is vertexpancyclic ordered、contains two edge-disjoint hamiltonian cycles、is panconnected),then L(G) also is1-hamiltonian(contains a 2-factor with k(k2) cycles、is vertexpancyclic ordered、contains two edge-disjoint hamiltonian cycles、is panconnected).