Projective Cubes
Let P2g+1 be the class of all planar signed graphs of type G11 of unbalanced-girth 2g+1.
Recall that to be a member of G11 is to say a cycle is unbalanced if and only if it is of odd-length, or that after a resigning if needed, all edges are negative.
Then a conjecture of Naserasr claims:
Conjecture. P2k+1 is bounded by SPC(2k).
Similarly let P2g be the class of all planar signed graphs of type G10 of unbalanced-girth at least 2g.
Recall that to be a member of G10 is to say all cycles are even, in other words, the underlying (planar) graphs is bipartite.
Then a conjecture of Guenin claims:
Conjecture. P2k is bounded by SPC(2k-1).
A more general question then is to ask:
Problem Given positive integers k and g satisfying k≤ g what are minimal subgraphs of SPC(2k) and SPC(2k-1) which bounds P2g+1 and P2g (respectively)?
The question is a common generalization of a number of theories in coloring planar graphs such the four-color theorem, the Grotzsch theorem, the fractional chromatic number of planar graphs of given odd-girth, or their circular chromatic number, the edge chromatic number of a class planar multigraphs and the Jaeger's conjecture on the theory of circular flows.