In this paper, we use the technique of the vertex insertion, considering the neighborhood intersections of the essential independent sets and the independent set of the partially square graphs on k-connected graphs G, to study almost Hamilton of graphs, and obtained three new sufficient conditions.

A Sufficient Condition for Hamilton K_1,m-free Graphs;

For a simple graph G=(V,E),Ore theorem states that G is Hamiltonian if every pair of nonadjacent vertices u and v satisfies d(u)+d(v)≥|V|.

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.

Moreover, the conjecture in the case of a block with cyclomatic number 2 is confirmed by constructing a Hamilton cycle.

Hamiltonicity,neighborhood union and square graphs of claw-free graphs;

Degree Sum of Independent Sets and Hamiltonicity of Graphs;

It is well-known that the problem on hamiltonicity of graphs is a very important problem in graph theory and the research is very active.

A New Sufficient Conditions and Hamiltonian graphs;

3 +e,P 4}, the following new localized result on hamiltonian graphs is obtained.