Rainbow Pancyclicity in a Collection of Graphs Under the Dirac-type Condition

Let G = {G_i: i ∈ [n]} be a collection of not necessarily distinct n-vertex graphs with the same vertex set V, where G can be seen as an edge-colored (multi)graph and each G_i is the set of edges with color i. A graph F on V is called rainbow if any two edges of F come from different G_is’. We say that G is rainbow pancyclic if there is a rainbow cycle C_ℓ of length ℓ in G for each integer ℓ ∈ [3, n]. In 2020, Joos and Kim proved a rainbow version of Dirac’s theorem: If \(\delta ({G_i}) \ge {n \over 2}\) $$\delta (G_i)\ge \frac{n}{2}$$ for each i ∈ [n], then there is a rainbow Hamiltonian cycle in G. In this paper, under the same condition, we show that G is rainbow pancyclic except that n is even and G consists of n copies of \({K_{{n \over 2},{n \over 2}}}\) $$K_{\frac{n}{2},\frac{n}{2}}$$ . This result supports the famous meta-conjecture posed by Bondy.