For graphs G and H , a homomorphism from G to H is a function φ : V (G) → V (H), which maps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H . Many cases of graph homomorphism and locally injective graph homomorphism are NPcomplete, so there is little hope to design polynomial-time...

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem i...

Let G be an almost linear Nash group, namely, a group that admits homomorphism with finite kernel to some GLk(R). A homology theory (the Schwartz homology) is established for the category of smooth Fréchet representations moderate growth. Frobenius reciprocity and Shapiro's lemma are proved in this category. As application, we give criterion automatic extensions homologies sections tempered G-v...

Abstract Let K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, for the existence a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ Weyl modules general linear group $GL_n(K)$ . Also, each p we so that corresponding spaces have dimension at least 2.

The signature of the Poincaré duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V ⊕ V ∗ where V is abstarct linear space with countable base. The space W is consider...

The oriented chromatic number of an oriented graph −→ G is the minimum order of an oriented graph −→ H such that −→ G admits a homomorphism to −→ H . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the m...

We study the complexity of structurally restricted homomorphism and constraint satisfaction problems. For every class of relational structures C, let LHOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B, when each element in A is only allowed a certain subset of elements of B as its image. We prove, under a certain complexity-theore...

In the classical Constraint Satisfaction Problem(CSP) two finite models are given and we are asked to find their homomorphism. In the Mixed-CSP problem, besides the models, a set of weighted pairs of elements of this two models is given and the task is to find a homomorphism that maximizes the weight of pairs consistent with the homomorphism, i.e. pairs for which homomorphism maps the first ele...