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Pré-Publication, Document De Travail Année : 2023

Packing signatures in signed graphs

Résumé

We define the signature packing number of a signed graph (G, σ), denoted ρ(G, σ), to be the maximum number of signatures σ 1 , σ 2 ,. .. , σ l such that each σ i is switching equivalent to σ and the sets E − σi , negative edges of (G, σ i), are pairwise disjoint. In this work, first in connection to recent developments on the theory of homomorphisms of signed graphs we prove that for a signed graph (G, σ), ρ(G, σ) ≥ d + 1 if and only if (G, σ) admits a homomorphism to SPC o d , where SPC o d is obtained from SPC d by adding a positive loop to every vertex. Noting that SPC d , signed projective cube of dimension d, is the signed (Cayley) graph built on Z d 2 where each pair of binary strings at hamming distance 1 are adjacent by a positive edge and those at hamming distance d are adjacent by a negative edge. In other words, SPC d is built from the hypercube of dimension d by considering all its edges as positive edges and adding a negative edge for each pair of antipodal vertices. In special cases we have: I. A simple graph G is 4-colorable if and only if ρ(G, −) ≥ 2. II. A signed bipartite graph (G, σ) maps to SP C 3 if and only if ρ(G, σ) ≥ 3 noting that SP C 3 is the same as (K 4,4 , M), that is a signed graph on K 4,4 where the set of negative edges forms a perfect matching. On restriction to planar graphs, I is then a restatement of the 4-color theorem and II is implied by an unpublished work of B. Guenin. After further development of this theory of packing in signed graphs, we give an independent proof of II which works on the larger class of K 5-minor-free graphs. More precisely we prove that: Theorem. If G is a K 5-minor-free bipartite simple graph, then for any signature σ we have ρ(G, σ) ≥ 4. The statement is shown to be strictly stronger than the four-color theorem and is proved assuming it. Furthermore, we show that I cannot be extended to the class of all signed planar simple graphs. Further development, including algorithmic implications, are considered.
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hal-03960731 , version 1 (27-01-2023)

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  • HAL Id : hal-03960731 , version 1

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Reza Naserasr, Weiqiang Yu. Packing signatures in signed graphs. 2023. ⟨hal-03960731⟩
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