



'''simple docstring'''





# using dfs for finding eulerian path traversal






def        lowerCamelCase  (				_snake_case  :						int     ,   _snake_case  :						Dict     ,   _snake_case  :						Optional[int]     ,   _snake_case  :						List[str]=None       ):





    '''simple docstring'''


    lowercase__		       =						(path or []) + [u]
    for v in graph[u]:
        if visited_edge[u][v] is False:
            lowercase__       ,       lowercase__		       =						True, True
            lowercase__		       =						dfs(lowerCAmelCase__     ,   lowerCAmelCase__     ,   lowerCAmelCase__     ,   lowerCAmelCase__       )
    return path




def        lowerCamelCase  (				_snake_case  :						int     ,   _snake_case  :						Tuple       ):





    '''simple docstring'''


    lowercase__		       =						0
    lowercase__		       =						-1
    for i in range(lowerCAmelCase__       ):
        if i not in graph.keys():
            continue
        if len(graph[i]       ) % 2 == 1:
            odd_degree_nodes += 1
            lowercase__		       =						i
    if odd_degree_nodes == 0:
        return 1, odd_node
    if odd_degree_nodes == 2:
        return 2, odd_node
    return 3, odd_node




def        lowerCamelCase  (				_snake_case  :						str     ,   _snake_case  :						List[str]       ):





    '''simple docstring'''


    lowercase__		       =						[[False for _ in range(max_node + 1       )] for _ in range(max_node + 1       )]
    lowercase__       ,       lowercase__		       =						check_circuit_or_path(lowerCAmelCase__     ,   lowerCAmelCase__       )
    if check == 3:
        print("graph is not Eulerian"       )
        print("no path"       )
        return
    lowercase__		       =						1
    if check == 2:
        lowercase__		       =						odd_node
        print("graph has a Euler path"       )
    if check == 1:
        print("graph has a Euler cycle"       )
    lowercase__		       =						dfs(lowerCAmelCase__     ,   lowerCAmelCase__     ,   lowerCAmelCase__       )
    print(lowerCAmelCase__       )




def        lowerCamelCase  (				):





    '''simple docstring'''


    lowercase__		       =						{1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
    lowercase__		       =						{1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
    lowercase__		       =						{1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
    lowercase__		       =						{1: [2, 3], 2: [1, 3], 3: [1, 2]}
    lowercase__		       =						{
        1: [],
        2: []
        # all degree is zero
    }
    lowercase__		       =						10
    check_euler(lowerCAmelCase__     ,   lowerCAmelCase__       )
    check_euler(lowerCAmelCase__     ,   lowerCAmelCase__       )
    check_euler(lowerCAmelCase__     ,   lowerCAmelCase__       )
    check_euler(lowerCAmelCase__     ,   lowerCAmelCase__       )
    check_euler(lowerCAmelCase__     ,   lowerCAmelCase__       )


if __name__ == "__main__":
 main()









'''simple docstring'''





def        lowerCamelCase  (				_snake_case  :						int     ,_snake_case  :						int       ):





    '''simple docstring'''


    return x if y == 0 else greatest_common_divisor(_snake_case     ,x % y       )




def        lowerCamelCase  (				_snake_case  :						int     ,_snake_case  :						int       ):





    '''simple docstring'''


    return (x * y) // greatest_common_divisor(_snake_case     ,_snake_case       )




def        lowerCamelCase  (				_snake_case  :						int = 20       ):





    '''simple docstring'''


    lowercase__		       =						1
    for i in range(1     ,n + 1       ):
        lowercase__		       =						lcm(_snake_case     ,_snake_case       )
    return g


if __name__ == "__main__":
 print(f'''{solution() = }''')




