Characterizing k-th powers of Hamilton cycles in G(n,W) via K_{k+1}-fractional covers
Establish that for every fixed integer k ≥ 2 and graphon W, the inhomogeneous random graph G(n,W) is asymptotically almost surely Hamiltonian to the k-th power if and only if: (1) W is connected; (2) the small-degree tail is light, i.e., lim_{alpha→0} μ({x ∈ Ω : deg_W(x) ≤ α})/α = 0; and (3) the only K_{k+1}-fractional cover of W with L1 norm at most 1/(k+1) is the constant function equal to 1/(k+1), where a K_{k+1}-fractional cover is a measurable f: Ω → [0,1] such that for μ^{k+1}-almost every (x1,…,x_{k+1}) either Σ_{i=1}^{k+1} f(x_i) ≥ 1 or at least one edge of the (k+1)-clique is absent, i.e., ∏_{1≤i<j≤k+1} W(x_i,x_j) = 0.
References
Hence, our conjecture, whose case k=1 corresponds to Theorem~\ref{thm:mainHC}, is as follows. Suppose that $W:\Omega2\to[0,1]$ is a graphon, and $k\ge 2$ is fixed. Then $G(n,W)$ contains a.a.s.\ the $k$-th power of a Hamilton cycle if and only if the following three conditions are fulfilled: (i) $W$ is a connected graphon, (ii) we have $\lim_{\alpha\searrow 0}\frac{\mu(D_\alpha(W))}\alpha=0$, and (iii) the only $K_{k+1}$-fractional cover of $W$ of $L1$-norm at most $\frac1{k+1}$ is the constant-$\frac{1}{k+1}$ function.