Naming a room for private and group chat

I am making a chat app. Below is the schema, which looks good to me. schema

These are the feature of this chat app that I need to implement.

  1. Group chat (room)
  2. Private chat: One to One

I am confused about what to name the group?

If person A sends a message to another person B. It should show B on A's screen and A on B's screen, like another chat app.

So I thought to name it A_B. But now the problem is I have to first split it and compare it with the requested user's name.

So, if user A requested I would split and send all the messages with group name B and vice versa.

It doesn't look right to me.

Also, the Same problem is with group chat.