Yes! Sorry, I mixed up two sentences in my head there. I meant to say that you can color 2ⁿ⁻¹ green and 2ⁿ⁻¹ red (or, alternatively, you can color all 2ⁿ vertic

Yes! Sorry, I mixed up two sentences in my head there. I meant to say that you can color 2ⁿ⁻¹ green and 2ⁿ⁻¹ red (or, alternatively, you can color all 2ⁿ vertices with one of green or red) such that no two vertices of the same color share an edge. (Which is what you describe!) It’s a bit late for me to edit now, but alas :)