Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections

Abstract

Let $\mathbf{G}=\{G_1,\dots ,G_{2n}\}$ be a collection of not necessarily distinct balanced bipartite graphs of order $2n$ with a common bipartition $V=(X,Y)$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\le 2n$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $\varphi :E(C)\to [2n]$ such that $e\in E(G_{\varphi (e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow bipancyclic} if it contains a rainbow cycle of every even length from 4 to $2n$. In this paper, we show that if $e(G_i)\ge n^2-n+1$ for each $i\in [2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\cdots =G_{2n}\cong B_n^1$, where $B_n^1$ is the graph obtained from $K_{n,n}$ by deleting all edges in its one subgraph $K_{n-1,1}$. As a corollary, we obtain that if $\rho (G_i)\ge \rho (B_n^1)>\sqrt{n(n-1)}$ for each $i\in [2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\cdots =G_{2n}\cong B_n^1$, which strengthens Theorem $5$ of [Electron. J. Combin., \textbf{33}(3)(2026), \#P3.29] and Theorem $1.6$ of [Linear Algebra Appl., \textbf{743}(2026), 250--270] from rainbow Hamiltonicity to rainbow bipancyclicity. We also characterize all bipartite graph collections that are not rainbow bipancyclic under the condition $\rho (G_i)\ge \sqrt{n(n-1)}$ for each $i\in [2n]$