Sorry, I know it's looked down upon to answer one's own question, but I figured out my mistake:
No graph that contains a 4-clique will ever be 3-colorable, but there are graphs that contain degree >= 3 that are 3-colorable.
Take this counter-example:
- 2 -
/ \
1 - 2 - 1
\ /
- 2 -
Notice how the "1"s have degree >= 3 but the graph itself is still 3-colorable.