If my father passed away 6 days ago and my mother is going into assisted living, is it legal to put their home inmy name?

It can be transferred to you, but your mother needs to make the transfer. There are different ways to do this: via a quitclaim deed, by selling it to for some reasonable or even nominal price, but putting it in trust with you as a beneficiiary, etc. etc. There are probably a half dozen or more ways to get it to you, but they each have different tax consequences for and/or your mom...as well as having different impacts under Medicare laws (if you mom is going to get Medicare) regarding the necessity of a person to contribute to their own care; an improper transfer can be invalidated or result in legal liability.

Consult with a trusts and estates attorney who can evaluate your situation and your goals and help determine the best way to accomplish your goals. Note that if your mother is mentally incompetent at this point, if you don't already have a power of attorney, you may also need to establish a guardianship over her.

I am so sorry for your loss and for your circumstances.  May I suggest that you seek help from an attorney in your area as soon as you can.  It is very difficult in this forum to discuss issues regarding assets without knowing what other assets your Mother has and her financial situation in general.  generally speaking, you should not transfer any of her assets without consulting n attorney.  If she can not pay for the assisted living I am sure that she is going to have to sign something that given them a lien on her assets.  Transferring them could be fraud.  If she is going in to the assisted living through a federal or state funded program then there is an automatic lien on the property as well.  Get help.  Good luck.