


# using dfs for finding eulerian path traversal
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__=None						):
 __lowerCamelCase		:							Any		 =						(path or []) + [u]
 for v in graph[u]:
  if visited_edge[u][v] is False:
   __lowerCamelCase ,					__lowerCamelCase		:							Union[str, Any]		 =						True, True
   __lowerCamelCase		:							Dict		 =						dfs(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 return path
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						):
 __lowerCamelCase		:							Tuple		 =						0
 __lowerCamelCase		:							int		 =						-1
 for i in range(SCREAMING_SNAKE_CASE__						):
  if i not in graph.keys():
   continue
  if len(graph[i]						) % 2 == 1:
   odd_degree_nodes += 1
   __lowerCamelCase		:							Dict		 =						i
 if odd_degree_nodes == 0:
  return 1, odd_node
 if odd_degree_nodes == 2:
  return 2, odd_node
 return 3, odd_node
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						):
 __lowerCamelCase		:							Any		 =						[[False for _ in range(max_node + 1						)] for _ in range(max_node + 1						)]
 __lowerCamelCase ,					__lowerCamelCase		:							List[Any]		 =						check_circuit_or_path(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 if check == 3:
  print('graph is not Eulerian'						)
  print('no path'						)
  return
 __lowerCamelCase		:							Union[str, Any]		 =						1
 if check == 2:
  __lowerCamelCase		:							Union[str, Any]		 =						odd_node
  print('graph has a Euler path'						)
 if check == 1:
  print('graph has a Euler cycle'						)
 __lowerCamelCase		:							Union[str, Any]		 =						dfs(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 print(SCREAMING_SNAKE_CASE__						)
def 				UpperCamelCase__					(  ):
 __lowerCamelCase		:							Optional[int]		 =						{1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
 __lowerCamelCase		:							str		 =						{1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
 __lowerCamelCase		:							Union[str, Any]		 =						{1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
 __lowerCamelCase		:							List[str]		 =						{1: [2, 3], 2: [1, 3], 3: [1, 2]}
 __lowerCamelCase		:							List[Any]		 =						{
     1: [],
     2: []
     # all degree is zero
 }
 __lowerCamelCase		:							str		 =						10
 check_euler(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 check_euler(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 check_euler(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 check_euler(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 check_euler(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)


if __name__ == "__main__":
       main()





from maths.is_square_free import is_square_free
from maths.prime_factors import prime_factors
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__						):
 __lowerCamelCase		:							Dict		 =						prime_factors(SCREAMING_SNAKE_CASE__						)
 if is_square_free(SCREAMING_SNAKE_CASE__						):
  return -1 if len(SCREAMING_SNAKE_CASE__						) % 2 else 1
 return 0


if __name__ == "__main__":
       import doctest

       doctest.testmod()

