[TECH] Algorithm to find euler circuit in a graph in O(|E|)

Important observation I found is that, if we delete a cycle (delete all the edges which form a cycle) from the graph which has a Euler circuit, the property of the graph still remains, i.e the necessary and sufficient condition for the graph to have a Euler cycle is that every vertex should have even degree. And also there has to be a cycle at every vertex.

INPUT: G =(E,V)
v = v
euler_cycle = NULL;
while(|E| > 0){
 /*Find a cycle in G, start
   the search at v, returns c0 around 
  which cycle is formed*/
 c0 = FIND_CYCLE(G,v);
 v = GET_NON_ZERO_DEGREE_VERTEX(c0); 
/*Above can be done in constant time,
 using some space while finding the cycle
 */
 MERGE(c0,euler_cyle);
/*constant time operation,
 put c0 after an edge e1 in
 euler cycle which ends at 
 vertex 'v', we can just remember 
 the position of v n euler_cycle
 every time we merge*/

}