Abstract
Let and be integers satisfying 1≤m≤(n-2)/2, and let [n]={1,...,n}. Let 
be a family of graphs (not necessarily distinct) defined on the common vertex set 
. A graph H on V is said to be rainbow if there exists a bijection 
such that every edge 
satisfies 
In this paper, we prove that if the signless Laplacian radius 
satisfies the inequality
For each 
, then 
contains a rainbow matching, except when all graphs in are identical and isomorphic to either 
or 
. Furthermore, for 
and 
, we show that if
for each 
, then 
admits a rainbow Hamilton path, unless all graphs in 
are identical and isomorphic to 
.
References
[1] Bradshaw P. Transversals and Bipancyclicity in Bipartite Graph Families. Electron J Combin. 2021;28(4):P4.25.
[2] Cheng Y, Sun W, Wang G, Wei L. Transversal Hamilton Paths and Cycles. arXiv preprint arXiv:2406.13998. 2024.
[3] Gupta P, Hamann F, Müyesser A, Parczyk O, Sgueglia A. A General Approach to Transversal Versions of Dirac-Type Theorems. Bull Lond Math Soc. 2023;55(6):2817-39.
[4] Joos F, Kim J. On a Rainbow Version of Dirac's Theorem. Bull Lond Math Soc. 2020;52(3):498-504.
[5] Li L, Li P, Li X. Rainbow Structures in a Collection of Graphs with Degree Conditions. J Graph Theory. 2023;104(2):341-59.
[6] Aharoni R, DeVos M, González Hermosillo de la Maza S, Montejano A, Šámal R. A Rainbow Version of Mantel's Theorem. Adv Comb. 2020;Paper No. 2:12pp.
[7] He Z, Frankl P, Győri E, Lv Z, Salia N, Tompkins C, et al. Extremal Results for Graphs Avoiding a Rainbow Subgraph. arXiv preprint arXiv:2204.07567. 2022.
[8] Keevash P, Saks M, Sudakov B, Verstraëte J. Multicolour Turán Problems. Adv Appl Math. 2004;33(2):238-62.
[9] Magnant C. Density of Gallai Multigraphs. Electron J Combin. 2015;22(1):1-28.
[10] Guo M, Lu H, Ma X, Ma X. Spectral Radius and Rainbow Matchings of Graphs. Linear Algebra Appl. 2023;679:30-7.
[11] He X, Li Y, Feng L. Spectral Radius and Rainbow Hamilton Paths of a Graph. Discrete Math. 2024;347(10):114128,11pp.
[12] Zhang Y, van Dam ER. Rainbow Hamiltonicity and the Spectral Radius. arXiv preprint arXiv:2401.17845. 2024.
[13] Sun W, Wang G, Wei L. Transversal Structures in Graph Systems: A Survey. arXiv preprint arXiv:2412.01121. 2024.
[14] Zhang L, Zhang Z. Spectral Radius and Rainbow k-Factors of Graphs. arXiv preprint arXiv:2501.08162. 2025.
[15] Brouwer A, Haemers W. Spectra of Graphs. New York: Springer Science & Business Media; 2011.
[16] Csikvári P. On a Conjecture of V. Nikiforov. Discrete Math. 2009;309:4522-6.
[17] Hong Y, Zhang X-D. Sharp Upper and Lower Bounds for Largest Eigenvalue of the Laplacian Matrices of Trees. Discrete Math. 2005;296:187-97.
[18] Yu G. On the Maximal Signless Laplacian Spectral Radius of Graphs with Given Matching Number. Proc Japan Acad Ser A. 2008;84:163-6.
[19] Huang H, Loh P, Sudakov B. The Size of a Hypergraph and Its Matching Number. Combin Probab Comput. 2012;21:442-50.
[20] Yu G-D, Fan Y-Z. Spectral Conditions for a Graph to Be Hamilton-Connected. Appl Mech Mater. 2013;336-338:2329-34.