((Contractible Edge of Eulerian Graph- Regular ))

Abstract

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 [7] 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 [7]. 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 [7]. 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.