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

Abstract

The packing number of a signed graph $(G, \sigma )$, denoted $\rho (G, \sigma )$, is the maximum number l of signatures $\sigma _1, \sigma _2,\ldots , \sigma _l$ such that each $\sigma _i$ is switching equivalent to $\sigma$ and the sets of negative edges $E^{-}_{\sigma _i}$ of $(G,\sigma _i)$ 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 Based on this unpublished work, and using the language of packing T-joins, proofs for girth 6, 7, and 8 are published.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.