Homomorphisms of sparse signed graphs

Clément Charpentier (1) , Reza Naserasr (2) , Eric Sopena (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale

The notion of homomorphism of signed graphs, introduced quite recently, provides better interplay with the notion of minor and is thus of high importance in graph coloring. A newer, but equivalent, definition of homomorphisms of signed graphs, proposed jointly by the authors of this paper and Tom Zaslavsky, leads to a basic no-homomorphism lemma. According to this definition, a signed graph (G, σ) admits a homomorphism to a signed graph (H, π) if there is a mapping φ from the vertices and edges of G to the vertices and edges of H (respectively) which preserves adjacencies, incidences, and signs of closed walks (i.e., the product of the sign of their edges). For ij = 00, 01, 10, 11, let g ij (G, σ) be the length of a shortest closed walk of (G, σ) which is, positive and of even length for ij = 00, positive and of odd length for ij = 01, negative and of even length for ij = 10, negative and of odd length for ij = 11. For each ij, if there is no closed walk of the corresponding type, we let g ij (G, σ) = ∞. If G is bipartite, then g 01 (G, σ) = g 11 (G, σ) = ∞. In this case, g 10 (G, σ) is certainly realized by a cycle of G, and it will be referred to as the unbalanced-girth of (G, σ).

Domaines: Mathématique discrète [cs.DM]

Clément Charpentier, Reza Naserasr, Eric Sopena. Homomorphisms of sparse signed graphs. The Electronic Journal of Combinatorics, 2020, 27 (3), ⟨10.37236/8478⟩. ⟨hal-03035030⟩