Edge and spectral conditions for rainbow pancyclicity in graph collections
Abstract
Let G={G1,…,Gn} be a collection of not necessarily distinct n-vertex graphs with a common vertex set V. A cycle C with V(C)⊆ V and |E(C)|≤ n is called \emph{rainbow} in G, if there exists an injection ϕ: E(C)→ [n] such that e∈ E(Gϕ(e)) for each e∈ E(C). The graph collection G is said to be \emph{rainbow pancyclic} if it contains a rainbow cycle of every length from 3 to n. In this paper, we show that if e(Gi)≥ binomn−12+1 for each i∈[n] with n≥ 3, then G is rainbow pancyclic, apart from three explicitly described exceptional graph collections. This answers Problem $1$ of [Discrete Math., \textbf {348}(2025), 114600] and strengthens the result from rainbow Hamiltonicity to rainbow pancyclicity. As a consequence, we obtain that if ρ(Gi)>n−2 for each i∈[n], then G is rainbow pancyclic unless G1=G2=…=Gn≅ K1∨(Kn−2∪ K1), which improves Theorem $5$ of [Discrete Math., \textbf {348}(2025), 114600]. We also characterize all graph collections that are not rainbow pancyclic under the condition ρ(Gi)≥ n−2 for each i∈[n].
Related
- Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections
- Local Ore Conditions for Rainbow Pancyclicity and Vertex-Pancyclicity in Graph Systems
- Rainbow Hamiltonian paths in graph collections under edge-size conditions
- A Bondy-type theorem for rainbow pancyclicity in graph systems
- Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems
BibTeX
@misc{you2026edge,
title = {{Edge and spectral conditions for rainbow pancyclicity in graph collections}},
author = {Lihua You and Xiaoxue Zhang and Xinghui Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23532},
eprint = {2609.23532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23532},
}
RIS
TY - UNPB
TI - Edge and spectral conditions for rainbow pancyclicity in graph collections
AU - Lihua You
AU - Xiaoxue Zhang
AU - Xinghui Zhao
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23532
UR - https://arxiv.org/abs/2609.23532
AB - Let G={G_1,…,G_n} be a collection of not necessarily distinct n-vertex graphs with a common vertex set V. A cycle C with V(C)⊆ V and |E(C)|≤ n is called \emph{rainbow} in G, if there exists an injection ϕ: E(C)→ [n] such that e∈ E(G_ϕ(e)) for each e∈ E(C). The graph collection G is said to be \emph{rainbow pancyclic} if it contains a rainbow cycle of every length from 3 to n. In this paper, we show that if e(G_i)≥ binomn−12+1 for each i∈[n] with n≥ 3, then G is rainbow pancyclic, apart from three explicitly described exceptional graph collections. This answers Problem $1$ of [Discrete Math., \textbf {348}(2025), 114600] and strengthens the result from rainbow Hamiltonicity to rainbow pancyclicity. As a consequence, we obtain that if ρ(G_i)>n−2 for each i∈[n], then G is rainbow pancyclic unless G_1=G_2=…=G_n≅ K_1∨(K_n−2∪ K_1), which improves Theorem $5$ of [Discrete Math., \textbf {348}(2025), 114600]. We also characterize all graph collections that are not rainbow pancyclic under the condition ρ(G_i)≥ n−2 for each i∈[n].
ER -