A note on packing spanning trees in graphs and bases in matroids

TL;DR

This paper characterizes graphs and matroids where edge connectivity equals the maximum number of edge-disjoint spanning structures, providing unique decompositions with implications for communication protocols.

Contribution

It offers a description and characterization of such special graphs and matroids, including their unique decompositions, extending to matroids with a focus on bases and cogirth.

Findings

Characterization of graphs with edge connectivity equal to the maximum number of edge-disjoint spanning trees.

Extension of these concepts to matroids with cogirth equal to the number of disjoint bases.

Identification of unique decompositions for such graphs and matroids.

Abstract

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.