Homomorphism complexes were introduced by Lovász to study topological obstructions to graph colorings. We show that folding in the second parameter of the homomorphism complex yields a homotopy equivalence. We study how the Mycielski construction changes the homotopy type of the homomorphism complex. We construct graphs showing that the topological bound obtained by odd cycles can be arbitraril...

On a closed symplectic surface Σ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Σ) to the cohomology group H(Σ;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or...

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking an input graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, allowed images, to fixed target $(H,\pi)$. The complexity similar without (corresponding all being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but ve...

For digraphs D and H, a homomorphism of D to H is a mapping f : V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) ∈ A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i ∈ V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is ∑ u∈V (D) cf(u)(u). The minimum cost homomorphism for a fixed digraph H, denoted by MinHOM(H), asks whether or not an input digraphD, wit...

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H . Hell and Nešetřil gave a complete characterisation of the complexity of deciding whether there is a homomorphism ...