No Appointment Letter, No Confirmation Letter,do I Still Give 15 Days Notice?

Dear Ms,
Do not worry about notice period unless you have signed any document ie Appointment letter or any undertaking that you will abide by rules and regulations, as prevalent .
They can not take legal action unless you have not returned any item which may have been given to you by them for use in performing your work/job.
So, as others have also told you, just relax.
Only you will not have experience letter and in case feedback is solicited from your previous Co, the same may not be in your favor.
Col.Suresh Rathi

an employee working for 11 months in organization and he refused to take the appointment letter or kept postponing and finally he was absconding and now he is claiming for Full and Final settlement and other dues but company don't want to settle his due or give any reliving letter to him because he is absconding employee now the employee makes an allegation that employer didn't give him appointment letter and not settling his due so in this case whom to be blamed is it organization or employee????

It is the duty of the employer to issue appointment letter. Unless you have documented that you have tried to serve appointment letter to him and he 'refused' the said letter, you are at fault. E.g. you have sent appointment letter to his address by speed/ registered post and it was returned by postal department. If you do not have any documents of serving him the appointment letter, then he has every right in raising the dispute of not issuing him the appointment letter.
There is no term as 'absconding' in labour law. If he is absenting/ not coming to office without leave and/ or permission, then he is guilty of unauthorised absence. For such act, disciplinary action may be initiated.
From the bare reading, it looks like the company is in back footing.