Rainbow Pancyclicity in Graph Systems

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