Density of C−4-critical signed graphs

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study graphs, we show that notion 4-coloring graphs and captured, through simple operations, by C−4. In particular, 4-color theorem equivalent to: Given a planar G, obtained from G replacing edge with path length 2 maps We prove that, except for one particular on 7 vertices 9 edges, any n must have at least ⌈4n3⌉ edges. Moreover, value n≥9 there exists either or ⌈4n3⌉+1 many As an application, conclude all girth 8 map Furthermore, example 6 which does not C−4, showing best possible disproving conjecture Naserasr, Rollova Sopena.