A cycle double cover (CDC),, of a graph is a collection of cycles in uch that every edge of belongs to exactly two cycles of . In 1979 , Seymour conjectured that every bridgeless graph has a CDC ( CDC conjecture ) . In C hapter 2 , we considered the oriented perfect path double cover of graphs. Maxov'a and Nesetril conjectured that every graph except two complete graphs and has an OPPDC ( OPPDC conjecture ). In C hapter 2, among some other results, we presented a complete proof for this fact. Moreover, we proved that the minimal counterexample to this conjecture is 2-connected and 3-edge-connected. . Every OCDC of a bridgeless graph of order with at most n-1 directed cycles is called an SOCDC . In Chapter 3, it is conjectured that every 2 -connected graph except and admits an SOCDC ( SOCDC conjecture ) .We studied graphs with SOCDC and obtained some properties of the minimal counterexample to this conjecture . The minimal counterexample to the SOCDC conjecture is 3-connected and it has no non-trivial edge cut of size 3 . In Chapter 4 , the -simultaneous edge coloring of graphs is studied . Moreover , the properties of the extremal counterexample to the above conjecture are investigated . Also , a relation between 2-simultaneous edge coloring of a graph with a cycle double cover with certain properties is shown .