On the packing number of antibalanced signed simple planar graphs of negative girth at least 5

The packing number of a signed graph $$(G, \sigma )$$ ( G , σ ) , denoted $$\rho (G, \sigma )$$ ρ ( G , σ ) , is the maximum number l of signatures $$\sigma _1, \sigma _2,\ldots , \sigma _l$$ σ 1 , σ 2 , … , σ l such that each $$\sigma _i$$ σ i is switching equivalent to $$\sigma $$ σ and the sets of negative edges $$E^{-}_{\sigma _i}$$ E σ i - of $$(G,\sigma _i)$$ ( G , σ i ) are pairwise disjoint. A signed graph packs if its packing number is equal to its negative girth. A reformulation of some well-known conjecture in extension of the 4-color theorem is that every antibalanced signed planar graph and every signed bipartite planar graph packs. On this class of signed planar graph the case when negative girth is 3 is equivalent to the 4-color theorem. For negative girth 4 and 5, based on the dual language of packing T-joins, a proof is claimed by B. Guenin in 2002, but never published. Based on this unpublished work, and using the language of packing T-joins, proofs for girth 6, 7, and 8 are published. We have recently provided a direct proof for girth 4 and in this work extend the technique to prove the case of girth 5.