Mixing is Hard for Triangle-Free Reflexive Graphs.

In the problem Mix ( H ) one is given a graph G and must decide if the Hom-graph Hom ( G , H ) is connected. We show that if H is a triangle-free reflexive graph with at least one cycle, Mix ( H ) is coNP-complete. The main part of this is a reduction to the problem NonFlat ( H ) for a simplicial complex H, in which one is given a simplicial complex G and must decide if there are any simplicial maps ϕ from G to H under which some 1-cycles of G maps to homologically nontrivial cycle of H. We show that for any reflexive graph H, if the clique complex H of H has a free, nontrivial homology group H 1 ( H ), then NonFlat ( H ) is NP-complete.