Rainbow Pancyclicity in Graph Systems

Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. edge $\{e_1,\ldots,e_n\}$ such $e_i\in E(G_i)$ for $1\leq i \leq n$. This can viewed as version well-known Dirac theorem. In this paper, we prove asymptotically by showing every $\varepsilon>0$, an integer $N>0$, when $n>N$ any $n$ (\frac{1}{2}+\varepsilon)n$, cycle. Our main tool is absorption technique. Additionally, $\delta(G_i)\geq \frac{n+1}{2}$ $i$, one find cycles length $3,\ldots,n-1$.