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PreprintCC BY 4.0arXiv

Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs

  • Zhiwei Guo
  • Nanxi Pan
  • Li Li
  • Yangyang Chen
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Abstract

Let cal G={G1,…,Gk} be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition (X,Y), where |X|=a, |Y|=b and 2≤ k≤ a≤ b. A \emph{rainbow matching} of cal G is a set of pairwise disjoint edges that can be chosen from distinct members of cal G. Denote by q(G) the signless Laplacian spectral radius of a graph G. In this paper, we prove that if q(Gi)≥ b+k−1 for each i∈{1,2,…,k}, then cal G admits a rainbow matching of size k unless G1=⋯=Gk≅ Kk−1,b∪Ka−k+1̅, and show that the threshold is sharp and attained by the exceptional collection. The condition is also extended to larger collections for a prescribed level t. In addition, we obtain a lower bound for the rainbow matching number in terms of the ordered signless Laplacian spectral radii of the members, and provide a stability version of the extremal characterization. In the proofs, we use the shifting technique and a quotient matrix arising from an equitable partition of a signless Laplacian matrix.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026signless,
  title = {{Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs}},
  author = {Zhiwei Guo and Nanxi Pan and Li Li and Yangyang Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.26712},
  eprint = {2608.26712},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26712},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs
AU  - Zhiwei Guo
AU  - Nanxi Pan
AU  - Li Li
AU  - Yangyang Chen
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.26712
UR  - https://arxiv.org/abs/2608.26712
AB  - Let cal G={G_1,…,G_k} be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition (X,Y), where |X|=a, |Y|=b and 2≤ k≤ a≤ b. A \emph{rainbow matching} of cal G is a set of pairwise disjoint edges that can be chosen from distinct members of cal G. Denote by q(G) the signless Laplacian spectral radius of a graph G. In this paper, we prove that if q(G_i)≥ b+k−1 for each i∈{1,2,…,k}, then cal G admits a rainbow matching of size k unless G_1=⋯=G_k≅ K_k−1,b∪K_a−k+1̅, and show that the threshold is sharp and attained by the exceptional collection. The condition is also extended to larger collections for a prescribed level t. In addition, we obtain a lower bound for the rainbow matching number in terms of the ordered signless Laplacian spectral radii of the members, and provide a stability version of the extremal characterization. In the proofs, we use the shifting technique and a quotient matrix arising from an equitable partition of a signless Laplacian matrix.
ER  -