Density of C−4-critical signed graphs

Abstract

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 each of its proper subgraphs does. To motivate the study of C−4-critical signed graphs, we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism 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 ⌈4n3⌉ edges. Moreover, we show that for each value of n≥9 there exists a C−4-critical signed graph on n vertices with either ⌈4n3⌉ or ⌈4n3⌉+1 many edges.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 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena.