Rainbow Pancyclicity in Graph Systems

Let G₁,...,Gₙ be graphs on the same vertex set of size n, each graph with minimum degree δ(Gᵢ)≥ n/2. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set ₁,...,eₙ\ such that eᵢ∈ E(Gᵢ) for 1≤ i ≤ 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 ε>0, there exists an integer $N>0$, such that when $n>N$ for any graphs G₁,...,Gₙ on the same vertex set of size n with δ(Gᵢ)≥ (1/2+ε)n, there exists a rainbow Hamiltonian cycle. Our main tool is the absorption technique. Additionally, we prove that with δ(Gᵢ)≥ n+1/2 for each i, one can find rainbow cycles of length $3,...,n-1$.