Homomorphisms of planar signed graphs to signed projective cubes

Projective-planar signed graphs and tangled signed graphs

A projective-planar signed graph has no two vertex-disjoint negative circles. We prove that every signed graph with no two vertex-disjoint negative circles and no balancing vertex is obtained by taking a projective-planar signed graph or a copy of −K5 and then taking 1, 2, and 3-sums with balanced signed graphs.

Homomorphisms of Signed Graphs

A signed graph [G,Σ] is a graph G together with an assignment of signs + and − to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 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 fl...

On cographic matroids and signed-graphic matroids

We prove that a connected cographic matroid of a graph G is the bias matroid of a signed graph Σ iff G imbeds in the projective plane. In the case that G is nonplanar, we also show that Σ must be the projective-planar dual signed graph of an actual imbedding of G in the projective plane. As a corollary we get that, if G1, . . . , G29 denote the 29 nonseparable forbidden minors for projective-pl...

Homomorphisms of signed planar graphs

Signed graphs are studied since the middle of the last century. Recently, the notion of homomorphism of signed graphs has been introduced since this notion captures a number of well known conjectures which can be reformulated using the definitions of signed homomorphism. In this paper, we introduce and study the properties of some target graphs for signed homomorphism. Using these properties, w...

A characterization of projective-planar signed graphs