Let me trace a -cycle: nodes A-B-C-D-E-A. Color A red. B must be blue. C must be red. D must be blue. E must be red.
But E connects to A, and both are red. Conflict. Now try a -cycle: A-B-C-D-A. Color A red. B blue. C red. D blue. D connects to A: blue to red. No conflict. The pattern: odd cycles always end with the same color on both ends. Even cycles alternate perfectly. This is why the odd cycle test works.