For any two graphs F and G, let hom(F,G) denote the number of homomorphisms F → G, that is, adjacency preserving maps V (F ) → V (G) (graphs may have loops but no multiple edges). We characterize Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary. Email: [email protected]. Research sponsored by OTKA Grant No. 67867. CWI and University of Amsterdam, Amsterdam, The Netherlands....

For a fixed graph H, we consider the H-Recoloring problem : given a graph G and two H-colorings of G, i.e., homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G,H) complex. For H = Kk this is the problem of finding paths between k-colorings, which was recently shown...

Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between homomorphism densities to a linear inequality. Inspired by the work of Freedman,...