Graph coloring to reduce the number of qubits in VQE

Question

I am reading through the following article: https://arxiv.org/abs/1312.2579 and I really struggle to understand anything of the section "D. The standard graph-coloring method" on page 7 and "E. The band-coloring method" on page 8 as well. I've been through plenty of articles that say they use this method, but I am having a hard time understanding anything that is done here. If someone here has any idea, it would be really appreciated if a walkthrough could be provided.