Mapping sparse signed graphs to (K_{2k} , M )

Reza Naserasr (1) , Riste Skrekovski , Zhouningxin Wang (2) , Rongxing Xu (2)

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

A homomorphism of a signed graph (G, σ) to (H, π) is a mapping of vertices and edges of G to (respectively) vertices and edges of H such that adjacencies, incidences and the product of signs of closed walks are preserved. Motivated by reformulations of the k-coloring problem in this language, and specially in connection with results on 3-coloring of planar graphs, such as Grötzsch's theorem, in this work we consider bounds on maximum average degree which are sufficient for mapping to the signed graph (K_{2k} , σ_m) (k ≥ 3) where σ m assigns to edges of a perfect matching the negative sign. For k = 3, we show that the maximum average degree strictly less than 14/5 is sufficient and that this bound is tight. For all values of k ≥ 4, we find the best maximum average degree bound to be 3. While the homomorphisms of signed graphs is relatively new subject, through the connection with the homomorphisms of 2-edge-colored graphs, which are largely studied, some earlier bounds are already given. In particular, it is implied from Theorem 2.5 of "Borodin, O. V., Kim, S.-J., Kostochka, A. V., and West, D. B., Homomorphisms from sparse graphs with large girth. J. Combin. Theory Ser. B (2004)" that if G is a graph of girth at least 7 and maximum average degree 28/11 , then for any signature σ the signed graph (G, σ) maps to (K_6 , σ_m). We discuss applications of our work to signed planar graphs and, among others, we propose questions similar to Steinberg's conjecture for the class of signed bipartite planar graphs.

Domaines: Mathématique discrète [cs.DM]

Reza Naserasr, Riste Skrekovski, Zhouningxin Wang, Rongxing Xu. Mapping sparse signed graphs to (K_{2k} , M ). 2021. ⟨hal-03109532⟩