((Contractible Edge of Eulerian Graph- Regular ))
Keywords:contractible edge eulerian graph, connected graph, eulerian graph. regular graph.
In this paper define the contractible edge eulerian graph that, let is a class of Eulerian graphs , the edge e in is called contractible edge eulerian graph if . The necessary conditions for Eulerian graphs to have contractible edge eulerian have been introduced, further, the even and odd contractible edge eulerian graph have been studied , we also define the contractible edge eulerian graph class, the edge e in G is satisfied property contraction is called contractible edge eulerian if . Tutte  proved every 3-connected graph non isomorphic to have 3-contractible and proved every 3-connected graph on more than four vertices contains an edge whose contraction yield a new 3-connected graph . We proved graph G is eulerian graph has contractible edge if non isomorphic to . How over every 4-connected graph on at least seven vertices can be reduced to smaller 4-connected graph by contraction one or two edge subsequently . Also we discussed the graph G is eulerian on at least seven vertices can be contraction and saved the properties of eulerian graph. Let be a regular graph and eulerian graph, the edges e in is called contractible regular-eulerian graph if is regular-eulerian grah, We discussed relation contraction of eulerian-regular graph then has contractible if if then has not contractible regular-eulerian.