发表机构
School of Mathematics and Computer Science, Yan’an University（延安大学数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二分图集合，利用移位技巧，在无符号拉普拉斯谱半径满足特定条件时，证明其存在大小为k的彩虹匹配，给出了该类彩虹匹配的谱条件
AI 中文摘要
设𝒢={G₁,…,Gₖ}是顶点二分划(X,Y)上的二分图集合（顶点二分划可重复），其中|X|=a，|Y|=b，k、a、b均为正整数。若存在一组两两不交的边，且任意两条边来自𝒢中不同的二分图，则称该集合𝒢存在彩虹匹配。记q(G)为二分图G的无符号拉普拉斯谱半径。本文利用移位技巧证明：若对每个Gᵢ∈𝒢={G₁,…,Gₖ}，都满足q(Gᵢ)≥b+k-1+√[(k-1)b]，且2≤k≤a≤b，则𝒢存在大小为k的彩虹匹配，除非G₁=G₂=…=G_k≅K_{k-1,b}∪overline{K_{a-k+1}}
英文摘要
Let ${\cal G}=\{G_1,\ldots,G_k\}$ be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition $(X,Y)$, where $|X|=a$, $|Y|=b$ and $2\le k\le a\le b$. 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 $q(G_i)\ge b+k-1$ for each $i\in\{1,2,\ldots,k\}$, then ${\cal G}$ admits a rainbow matching of size $k$ unless $G_1=\cdots=G_k\cong K_{k-1,b}\cup\overline{K_{a-k+1}}$, 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.
