A signed graph [G, ] is a graph G together with an assignment of signs + and − to all the edges of G where is the set of negative edges. Furthermore [G, 1 ] and [G, 2 ] are considered to be equivalent if the symmetric difference of 1 and 2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph homomorphisms where he showed how

We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g − 1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs.

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient q∈R such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most q·|E(G)|. We show that if G is bridgeless, then q≤9, and in the general case q≤11.

Let G be a regular bipartite graph and X⊆E(G). We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph (G,X), that is a graph G with exactly the edges from X being negative, is not equivalent to (G,∅). In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.

Bouchet's conjecture asserts that each signed graph which admits a nowhere‐zero flow has a nowhere‐zero 6‐flow. We verify this conjecture for two basic classes of signed graphs—signed complete and signed complete bipartite graphs by proving that each such flow‐admissible graph admits a nowhere‐zero 4‐flow and we characterise those which have a nowhere‐zero 2‐flow and a nowhere‐zero 3‐flow.