HOSIGRA

Let P_2g+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".

A conjecture of Naserasr claims:

Conjecture. P_2k+1 is bounded by SPC(2k).

Observer that the first case is the four-color theorem.

Similarly let P_2g 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".

A conjecture of Guenin claims:

Conjecture. P_2k is bounded by SPC(2k-1).

A more general question proposed in "R. Naserasr. Mapping planar graphs into projective cubes. J. Graph theory, 74(3):249–259, 2013." 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 P_2g+1 and P_2g (respectively)?

The question is a common generalization of a number of theories in coloring planar graphs such as the four-color theorem, the Grötzsch theorem, the fractional chromatic number of planar graphs of given odd-girth, their circular chromatic number, the edge chromatic number of a class planar multigraphs and the Jaeger's conjecture on the theory of circular flows.