Edge-disjoint spanning trees and balloons in (multi-)graphs from size or spectral radius

A multigraph is a graph that may have multiple edges, but has no loops. The multiplicity of a multigraph is the maximum number of edges between any pair of vertices. The spanning tree packing number of a graph G, denoted by τ(G), is the maximum number of edge-disjoint spanning trees contained in G. A balloon of a graph G is a maximal 2-edge-connected subgraph that is joined to the rest of G by exactly one cut edge. By $b(G)$, $e(G)$, and κ(G), we denote the number of balloons, the size, and the vertex-connectivity of G, respectively. In this paper, we show that for a positive integer k and any multigraph G of order n≥ 2r with multiplicity m≤ k and minimum degree δ ≥2k, if e(G)≥ m[r2+n-r2]+k, then τ(G)≥ k, where r=(δ+1)/m. This extends the result of Fan, Gu and Lin (J. Graph Theory, 2023). Analogous results involving the size to characterize κ(G)≥ k or b(G)≤ k-1 of a multigraph G are also presented. In addition, we prove a tight sufficient condition to guarantee b(G)≤ k-1 in terms of the spectral radius of a simple graph G, with extremal graphs characterized.