Generalizing the 4-color theorem, perhaps one of the most motivating conjectures of graphs theory:
Conjecture. If every cycle of a planar signed graph \( (G, \sigma) \) maps to \( SPC(k) \), then \( (G, \sigma) \) itself maps to \( SPC(k) \).
Recall: \( SPC(k) \) is the signed projective cube of dimension \( k \). That is the Cayley signed graph whose vertices are the element of the binary group \( \mathbb{Z}_2^k \) where a pair of vertices at Hamming distance 1 are connected by a positive edge and those at hamming distance \( k \) are connected by a negative edge.
Special case: \( SPC(2) \) is switching equivalent to \( (K_4, -) \), that is \( K_4 \) with all edges negative. If we consider a planar signed graphs \( G, - \), then every cycle of it maps to \( (K_4, -) \) 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.