Edge-disjoint spanning trees and eigenvalues of regular graphs

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of k edge-disjoint spanning trees in a regular graph, when k∈{2,3}. More precisely, we show that if the second largest eigenvalue of a d-regular graph G is less than , then G contains at least k edge-disjoint spanning trees, when k∈{2,3}. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any k