In this paper,it is proved that convex n-polygon has some subdivision graphs are(n_1,n_2)-Euler graph,to any partition(n_1,n_2) of n,where n_1+n_2=n,n_2≡n_1(mod3),n_1≥0,n_2≥3.

It is proved that if G=(p,q) is a Euler graph,then J(G) is a Euler graph if and only if q is a singular number and q≥5,Also,let G=(p,q) is a connected graph,then J(G) is a Euler graph if and only if q≥5 is a singular number,q>ζ+1,and for v∈V(G),there is same parity for d(v) or q≥6 is a even number and(q>ζ+1),and for uv∈E(G),d(u),d(v) have different parity,there ζ=max{d(u)+d(v)｜uv∈E(G)}.

In this paper,we prove that non-planar Euler graph may be expressed as the union of cycles in which less than |V|-2 edges are disjoint under the certain condictions of the edge connected degree satisfied, of which |V|is the number of all vertices of the graph.

There is a theorem for judging supereulerian graph:let G be a z_edge_connected triangle_free simple graph on n≥31 vertices, if δ(G)≥n/10 , and G can t be contracted to K 2,3 ,then G has a spanning eulerian subgraph.

The collection of all supereulerian graphs will be denoted by SL.

G is a supereulerian graph.

The Determination of Supereulerian Graphs and Research of Catlin-Conjecture;

On contraction method to determine whether a graph is supereulerian;

In studying supereulerian graphs,estimating edge-number is a very interesting problem.

A graph is supereulerian if it has a spanning Eulerian subgraph.