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.

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.

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

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.

Matching up to the three indispensable conditions to conversion inference of categorical proposition in traditional logic is in fact proved to be the restriction of interrelation between lexical items that the conversion inference concerns, as is visually cleared by Eulerian graph.