Make the file

[root@risby test]# whoami
root
[root@risby test]# chmod 777 .
[root@risby test]# touch bar foo
[root@risby test]# chmod +t -v foo
mode of `foo' changed to 1644 (rw-r--r-T)
[root@risby test]# ls -al
total 8
drwxrwxrwx. 2 root root 4096 Jan  7 09:32 .
drwxr-xr-x. 9 root root 4096 Jan  7 09:20 ..
-rw-r--r--. 1 root root    0 Jan  7 09:32 bar
-rw-r--r-T. 1 root root    0 Jan  7 09:32 foo
[root@risby test]# 

Delete the file

[guest@risby test]$ whoami
guest
[guest@risby test]$ touch baz
[guest@risby test]$ ls -al
total 8
drwxrwxrwx. 2 root  root  4096 Jan  7 09:33 .
drwxr-xr-x. 9 root  root  4096 Jan  7 09:20 ..
-rw-r--r--. 1 root  root     0 Jan  7 09:32 bar
-rw-rw-r--. 1 guest guest    0 Jan  7 09:33 baz
-rw-r--r-T. 1 root  root     0 Jan  7 09:32 foo
[guest@risby test]$ rm bar
rm: remove write-protected regular empty file `bar'? y
[guest@risby test]$ rm foo
rm: remove write-protected regular empty file `foo'? y
[guest@risby test]$ ls -al
total 8
drwxrwxrwx. 2 root  root  4096 Jan  7 09:34 .
drwxr-xr-x. 9 root  root  4096 Jan  7 09:20 ..
-rw-rw-r--. 1 guest guest    0 Jan  7 09:33 baz
[guest@risby test]$ 
Note specifically that I am equally able to delete files with and without the sticky bit set, as long as I have write permission on the directory in which they reside.

The touch baz above is to show that I can write into that mode-777 directory.