Average minimum-degree conjecture for transversal cycle-factors

Let $C_k$ be the cycle on $k$ vertices, and let the $n$-blow-up of $C_k$ have parts $V_1,\dots,V_k$, each of size $n$. For a spanning subgraph $G$, let $\delta(G[V_i,V_{i+1}])$ denote the minimum degree in the bipartite graph between consecutive parts, with indices taken modulo $k$. A transversal $C_k$-factor is a collection of $n$ vertex-disjoint copies of $C_k$, each using one vertex from every part.

Average minimum-degree conjecture. For every $k\geq3$ and $\varepsilon>0$, there exists $n_0=n_0(k,\varepsilon)$ such that for every $n\geq n_0$, if there are numbers $\delta_1,\dots,\delta_k\geq n/2$ satisfying

$$\delta(G[V_i,V_{i+1}])\geq\delta_i\quad\text{for every }i\in[k]$$

and

$$\frac{1}{k}\sum_{i\in[k]}\delta_i\geq\left(1+\frac{1}{k}+\varepsilon\right)\frac{n}{2},$$

then $G$ has a transversal $C_k$-factor.

This conjecture strengthens the paper's asymptotic theorem by replacing a uniform degree condition with an average condition across the consecutive pairs. The preceding example shows that the corresponding threshold is close to best possible.