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 H₁(H), then NonFlat(H) is NP-complete.