# co.combinatorics – Can cycle homomorphisms dominate cycle subgraphs in dense enough graphs?

Let $$k$$ be a fixed positive integer and let $$G$$ be an $$n$$-node graph with average degree $$d ge C_k n^{1/k}$$. It is known that (if $$C_k$$ is large enough) there are $$Omega_k(d^{2k})$$ homomorphisms from a $$2k$$-cycle into $$G$$, and even that there are $$Omega_k(d^{2k})$$ instances of $$C_{2k}$$ as a subgraph, and that both of these lower bounds are best possible save the hidden constant.

Thus, on graphs where these lower bounds are both tight, a constant fraction of the homomorphisms are injections, i.e. correspond to subgraphs. My question is whether this is true of all graphs: if $$G$$ satisfies the above density condition, does this necessarily mean that a constant fraction $$c_k > 0$$ of its cycle homomorphisms are injective? I would guess that the answer is yes, but I haven’t been able to find this in prior work.