A.B. ATTAR - Department of Mathematics University of thi-qar collage of education for pure sciences

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles have been introduced. Further, the even and odd removable cycles in Eulerian graphs have also been studied. The necessary and sufficient conditions for regular graphs (digraphs) to have a removable cycles have been characterized. We also define, the removable cycle class.

Removable cycle, Connected graph, Eulerian graph