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HOSIGRA is a project to develop the theory of **homomorphisms of signed graphs** supported by L'Agence nationale de la recherche for the period of Feb 2018--Feb 2022.

This is a collaborative project among graph theory teams from IRIF, LaBRI and LIRMM. Therefore CNRS, and universities of Bordeaux, Montpellier and Paris Diderot are the main hosts. However the project will be carried on in collaboration with all interested parties.

**About the picture on the heading of the website**: there are two 3-dimensional hypercubes, one on the right another on the left. When a matching connecting corresponding vertices is added we get a 4-dimensional hypercube. An example of antipodal pair of vertices in the 4-dimensional cube is given in circled pair of vertices. A **projection** of the cube is to identify such antipodal pairs of vertices, thus projecting the right copy to the one on the left. In doing so edges of the right copy maps to edges on the left copy. However, edges connecting corresponding vertices of two copies will now become new edges connecting each vertex on the left copy to its antipodal on 3-dimensional cube of the left copy. One example of such connection is depicted by dotted black line.

With regard to colors of the edges: they represent presentation of the resulting graph as the Cayley graph:

- (
**ℤ**_{2}^{4}, {e_{1}, e_{2}, e_{3}, e_{4}, J}).

To obtain what would we call **signed projective cubes of dimension 4** (dimension k in general) one may take any of the color classes as the set of negative edges. This signed graphs is denoted by **SPC(4)** (**SPC(k)** in general).

Projective cubes are among most symmetric graphs. The example of this picture, the projective cube of dimension 4, is among most well-known graphs and it appears in a number of independent subjects.

Homomorphisms to signed projective cube are what motivated B. Guenin to define the notion of homomorphisms of singed graphs which is now the central subject of this project.