By extending the notion of indicator to signed graphs, we show the problem of 4-coloring is captured by the problem of C_{-4}-coloring (mapping signed graphs to a negative 4-cycle). Moreover, we extend the notion of H-critical to (H, \pi)-critical. In this talk, we prove that any C_{-4}-critical signed graph on n vertices, except for one particular signed bipartite graph on 7 vertices and 9 edges, must have at least 4n/3 edges. As an application to planarity, we conclude that every signed bipartite planar graph of negative girth at least 8 admits a homomorphism to C_{-4} and show that this bound is the best possible, which is relevant to a bipartite analog of Jaeger-Zhang conjecture.

I will give an overview of the following result. Let k and d be integers. Let G be a graph with maximum average degree at most 2k + 2d/(k+d+1). Then G decomposes into k+1 pseudoforests such that one of the pseudoforests has each connected component containing at most d edges. Here a pseudoforest is a graph where each component contains at most one cycle, and a decomposition is a partition of the edge set into edge disjoint subgraphs. By a result of Fan et al., the bound is best possible for all k and d, even when you ask for one pseudoforest with maximum degree at most d. This is joint work with Logan Grout.