On the packing number of antibalanced signed simple planar graphs of negative girth at least 5 - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2023

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

Résumé

The packing number of a signed graph (G, σ), denoted ρ(G, σ), is the maximum number of signatures σ 1 , σ 2 , • • • , σ l such that each σ i is switching equivalent to σ and the sets of negative edges E − σ i of (G, σ i) are pairwise disjoint. A signed graph packs if its packing number is equal to its negative girth. A reformulation of some wellknown 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 Based on this unpublished work, and using the language of packing T-joins, proofs for girth 6, 7, and 8 are published.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.
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Dates et versions

hal-04080407 , version 1 (24-04-2023)

Identifiants

  • HAL Id : hal-04080407 , version 1

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Reza Naserasr, Weiqiang Yu. On the packing number of antibalanced signed simple planar graphs of negative girth at least 5. 2023. ⟨hal-04080407⟩
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