Many edge-disjoint rainbow spanning trees in general graphs

A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for \(n\) and \(C\) large enough, if \(G\) is an edge-colored copy of \(K_n\) in which each color class has size at most \(n/2\), then \(G\) has at least \(\lfloor n/(C\log n)\rfloor\) edge-disjoint rainbow spanning trees. Here we strengthen this result by showing that if \(G\) is any edge-colored graph with \(n\) vertices in which each color appears on at most \(\delta\cdot\lambda_1/2\) edges, where \(\delta\geq C\log n\) for \(n\) and \(C\) sufficiently large and \(\lambda_1\) is the second-smallest eigenvalue of the normalized Laplacian matrix of \(G\), then \(G\) contains at least \(\left\lfloor\frac{\delta\cdot\lambda_1}{C\log n}\right\rfloor\) edge-disjoint rainbow spanning trees.