Generalizing the 4-color theorem, perhaps one of the most motivating conjectures of graphs theory:
Conjecture. If every cycle of a planar signed graph \( \Large (G, \sigma) \) maps to \( \Large SPC(k) \), then \( \Large (G, \sigma) \) itself maps to \( \Large SPC(k) \).
Recall: \( \Large SPC(k) \) is the signed projective cube of dimension \( \Large k \). That is the Cayley signed graph whose vertices are the element of the binary group \( \Large \mathbb{Z}_2^k \) where a pair of vertices at Hamming distance 1 are connected by a positive edge and those at hamming distance \( \Large k \) are connected by a negative edge.
Special case: \( \Large SPC(2) \) is switching equivalent to \( \Large (K_4, -) \), that is \( \Large K_4 \) with all edges negative. If we consider a planar signed graphs \( \Large G, - \), then every cycle of it maps to \( \Large (K_4, -) \) unless it has a loop. Thus this case of the conjecture is equivalent to claiming that any planar graph with no loop is 4-colorable.
Click to see a depiction of \( \Large SPC(1) \), \( \Large SPC(2) \), \( \Large SPC(3) \), and \( \Large SPC(4) \) with induction construction.