Density of C −4 -critical signed graphs

Reza Naserasr (1) , Lan Anh Pham (2) , Zhouningxin Wang (2)

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

A signed bipartite (simple) graph (G, σ) is said to be C −4-critical if it admits no homomorphism to C −4 (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomor-phism to C −4. In particular, the 4-color theorem is equivalent to: Given a planar graph G, the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to C −4. We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any C −4-critical signed graph on n vertices must have at least 4n 3 edges, and that this bound or 4n 3 + 1 is attained for each value of n ≥ 9. As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to C −4. Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to C −4 , showing that 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the above mentioned restatement of the 4CT.

Domaines: Mathématique discrète [cs.DM]

Reza Naserasr, Lan Anh Pham, Zhouningxin Wang. Density of C −4 -critical signed graphs. Journal of Combinatorial Theory, Series B, 2022. ⟨hal-03000545⟩