Special case of Rota's basis conjecture on graphic matroids (Q2088711)

Summary: Gian-Carlo Rota conjectured that for any \(n\) bases \(B_1, B_2, \ldots, B_n\) in a matroid of rank \(n\), there exist \(n\) disjoint transversal bases of \(B_1, B_2, \ldots, B_n\). The conjecture for graphic matroids corresponds to the problem of an edge-decomposition as follows; If an edge-colored connected multigraph \(G\) has \(n-1\) colors and the graph induced by the edges colored with \(c\) is a spanning tree for each color \(c\), then \(G\) has \(n-1\) mutually edge-disjoint rainbow spanning trees. In this paper, we prove that edge-colored graphs where the edges colored with \(c\) induce a spanning star for each color \(c\) can be decomposed into rainbow spanning trees.