Spectral radius conditions for edge-disjoint spanning trees in $(k+c)$-edge-connected graphs

TL;DR

This paper establishes tight spectral radius conditions for $(k+c)$-edge-connected graphs to contain $k$ edge-disjoint spanning trees, extending previous results and characterizing extremal graph families.

Contribution

It generalizes spectral radius conditions for edge-disjoint spanning trees to all $(k+c)$-edge-connected graphs, identifying extremal graphs and their structures.

Findings

Derived tight spectral radius bounds for $(k+c)$-edge-connected graphs.

Characterized extremal graphs with maximum spectral radius.

Identified graph families with specific clique and remaining part structures.

Abstract

Let $\tau(G)$ denote the spanning tree packing number of a graph $G$. Recently, Zhang and Fan [J. Graph Theory 112 (2) (2026) 128--144] posed the problem of finding a tight spectral radius condition for an $m$-edge-connected graph $G$ to guarantee $\tau(G)\ge k$ for $k+1\le m\le 2k-1$. They solved the cases $m=k$ and $k=2, m=3$. In this paper, we study this problem for all $m=k+c$, where $1\le c\le k-1$. For $1\le c\le k-2$, we obtain a tight spectral radius condition for a $(k+c)$-edge-connected graph to contain $k$ edge-disjoint spanning trees. We also obtain a tight spectral radius condition for $(2k-1)$-edge-connected graphs. In both cases, we give graph families containing all extremal graphs, and the graphs with maximum spectral radius in these families serve as the corresponding extremal graphs. Each graph in these families consists of a large clique and a small remaining part, with certain restrictions on the edges inside the small part and between the two parts. Moreover, for the case $m=k+1$, we further determine the unique extremal graph.