Special Case of Rota’s Basis Conjecture on Graphic Matroids - Maezawa, Shun-ichi; Yazawa, Akiko (The Electronic Journal of Combinatorics, 2022)
Gian-Carlo Rota conjectured that for any n bases B_1,B_2,…,B_n in a matroid of rank n, there exist n disjoint transversal bases of B_1,B_2,…,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.