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 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 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 ⌈ 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 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the above mentioned restatement of the 4CT.

Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

“…A classic relation between the chromatic number of a graph and homomorphism from a certain subdivisions of it to the odd cycle is extended, in [19], to a relation between the circular chromatic number of signed graphs and homomorphism of its subdivision to negative cycles. Here we present a slightly stronger version and then use it to build examples in the next sections.…”

“…Indeed that is the case for (K 4,4 , M ): it bounds the class of all signed bipartite planar simple graphs [20]. As the limit of the circular chromatic number of signed bipartite planar simple graphs is 4 (see [23] and [12]), this cannot be the It is shown in [19] that every signed bipartite planar graph of negative girth at least 8 maps to C −4 and that this girth condition cannot be improved to 6. We observe that C −4 is a subgraph of each of the three homomorphism targets of this discussion.…”

“…A bipartite analogue of this conjecture was first proposed in [21], but considering the result of [19], it is modified to the following.…”

Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

“…[Jaeger-Zhang Conjecture] Given a positive integer p, we have χ c (P * 4p+1 ) ≤ 2p+1 p . A bipartite analogue of this conjecture was first proposed in [21], but considering the result of [19], it is modified to the following.…”