Sure! The problem you've provided involves generating test cases for a string recovery problem based on given constraints. Below is a Python function to generate test cases for this problem:

```python
import random
import string

def generate_string(length):
    """Generate a random string of given length consisting of lowercase and uppercase letters."""
    return ''.join(random.choices(string.ascii_letters, k=length))

def tcgen():
    # Random length for the string, constrained by problem limits
    l = random.randint(1, 100000)
    # Random acceptable Hamming distance, constrained by problem limits
    d = random.randint(0, min(5000, l))
    
    # Three strings of length `l`
    str_1 = generate_string(l)
    str_2 = generate_string(l)
    str_3 = generate_string(l)
    
    return l, d, str_1, str_2, str_3

# Example of generating a single test case
l, d, str_1, str_2, str_3 = tcgen()
print(f"{l} {d}")
print(str_1)
print(str_2)
print(str_3)
```

### Explanation:

1. **generate_string(length)**:
   - This function creates a random string of a specified length using lowercase and uppercase alphabets.

2. **tcgen()**:
   - The function generates the variables according to the constraints provided in the problem statement:
     - `l` is a random integer between 1 and 100,000.
     - `d` is a random integer between 0 and the minimum of 5,000 or `l`.
     - Three random strings `str_1`, `str_2`, and `str_3` each of length `l`.

3. **Output**:
   - The function returns the generated values which represent a test case.
   - Example usage generates and prints one test case.

This generator will help create diverse test cases that adhere to the constraints and requirements specified in the problem statement. You can adjust the function to produce multiple test cases or manipulate the randomness to target specific edge cases if needed.