Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections
Abstract
Let G={G1,…,G2n} 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)⊆ V and |E(C)|≤ 2n is called \emph{rainbow} in G, if there exists an injection ϕ: E(C)→ [2n] such that e∈ E(Gϕ(e)) for each e∈ E(C). The graph collection 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(Gi)≥ n2−n+1 for each i∈[2n], then G is rainbow bipancyclic unless G1=G2=…=G2n≅ Bn1, where Bn1 is the graph obtained from Kn,n by deleting all edges in its one subgraph Kn−1,1. As a corollary, we obtain that if ρ(Gi)≥ ρ(Bn1)>√(n(n−1)) for each i∈[2n], then G is rainbow bipancyclic unless G1=G2=…=G2n≅ Bn1, 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 ρ(Gi)≥ √(n(n−1)) for each i∈[2n]
Related
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- Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs
- Rainbow Hamiltonian paths in graph collections under edge-size conditions
- Spectral radius and size conditions for 1-tough balanced bipartite graphs to contain a k-factor with k≥ 3
- Local Ore Conditions for Rainbow Pancyclicity and Vertex-Pancyclicity in Graph Systems
BibTeX
@misc{you2026edge,
title = {{Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections}},
author = {Lihua You and Xiaoxue Zhang and Xinghui Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16459},
eprint = {2610.16459},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16459},
}
RIS
TY - UNPB
TI - Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections
AU - Lihua You
AU - Xiaoxue Zhang
AU - Xinghui Zhao
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16459
UR - https://arxiv.org/abs/2610.16459
AB - Let G={G_1,…,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)⊆ V and |E(C)|≤ 2n is called \emph{rainbow} in G, if there exists an injection ϕ: E(C)→ [2n] such that e∈ E(G_ϕ(e)) for each e∈ E(C). The graph collection 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)≥ n^2−n+1 for each i∈[2n], then G is rainbow bipancyclic unless G_1=G_2=…=G_2n≅ 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 ρ(G_i)≥ ρ(B_n^1)>√(n(n−1)) for each i∈[2n], then G is rainbow bipancyclic unless G_1=G_2=…=G_2n≅ 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 ρ(G_i)≥ √(n(n−1)) for each i∈[2n]
ER -