Many edge-disjoint rainbow spanning trees in general graphs

Abstract

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.