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Two-disjoint-cycle-cover edge bipancyclicity in Cartesian products
Abstract
For two even integers r1≤ r2, a graph G of even order is \emph{2-DCC edge [r1,r2]-bipancyclic} if for every ordered pair (e,f) of independent edges and every even integer ℓ∈[r1,r2], there exist two vertex-disjoint cycles C1 and C2 with V(C1)∪ V(C2)=V(G), |C1|=ℓ, |C2|=|V(G)|−ℓ, e∈ E(C1), and f∈ E(C2). We study this property for Cartesian products G=H1□⋯□ Hd with d≥2, where each factor is K2 or an even-order bipartite edge-Hamiltonian graph. We prove that G is 2-DCC edge $[4,|V(G)|/2]$-bipancyclic whenever d≥4, and also for d∈{2,3} under explicit Hamiltonian-cycle conditions on Hi. Here 1≤ i≤ d. As applications, we give sharp characterizations for undirected toroidal meshes, star-cube networks, and hyperstar networks.
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BibTeX
@misc{lu2026twodisjointcyclecover,
title = {{Two-disjoint-cycle-cover edge bipancyclicity in Cartesian products}},
author = {Ke Lu and Ruichao Niu and Min Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23031},
eprint = {2609.23031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23031},
}
RIS
TY - UNPB
TI - Two-disjoint-cycle-cover edge bipancyclicity in Cartesian products
AU - Ke Lu
AU - Ruichao Niu
AU - Min Xu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.23031
UR - https://arxiv.org/abs/2609.23031
AB - For two even integers r_1≤ r_2, a graph G of even order is \emph{2-DCC edge [r_1,r_2]-bipancyclic} if for every ordered pair (e,f) of independent edges and every even integer ℓ∈[r_1,r_2], there exist two vertex-disjoint cycles C_1 and C_2 with V(C_1)∪ V(C_2)=V(G), |C_1|=ℓ, |C_2|=|V(G)|−ℓ, e∈ E(C_1), and f∈ E(C_2). We study this property for Cartesian products G=H_1□⋯□ H_d with d≥2, where each factor is K_2 or an even-order bipartite edge-Hamiltonian graph. We prove that G is 2-DCC edge $[4,|V(G)|/2]$-bipancyclic whenever d≥4, and also for d∈{2,3} under explicit Hamiltonian-cycle conditions on H_i. Here 1≤ i≤ d. As applications, we give sharp characterizations for undirected toroidal meshes, star-cube networks, and hyperstar networks.
ER -