Eigenvalue conditions implying edge-disjoint spanning trees and a forest with constraints
Jin Cai, Bo Zhou
TL;DR
This work links spectral graph theory to spanning-tree packing by establishing two sharp eigenvalue conditions that enforce a robust edge-disjoint spanning-tree structure together with a large auxiliary forest. It defines property P(k,δ) to capture the dual objective of τ(G) ≥ k and a surplus forest, and proves (i) a spectral radius criterion λ1(G) ≥ λ1(B_{n,δ+1}^{k−1}) for δ ≥ 2k+2 and n ≥ 2δ+3, unless G is the extremal graph B_{n,δ+1}^{k−1}, and (ii) a second-largest eigenvalue criterion λ2(G) < δ − 2(k+(δ−1)/δ)/(δ+1) ensuring P(k,δ). The proofs synthesize the Tutte–Nash-Williams tree packing framework with interlacing and quotient-matrix techniques, along with sharp subgraph spectral bounds, and extend to an α-parameter family via λ_{α,2}. The results yield tight, practically verifiable spectral criteria for guaranteeing multiple edge-disjoint spanning trees and a substantial accompanying forest.
Abstract
Let $G$ be a nontrivial graph with minimum degree $δ$ and $k$ an integer with $k\ge 2$. In the literature, there are eigenvalue conditions that imply $G$ contains $k$ edge-disjoint spanning trees. We give eigenvalue conditions that imply $G$ contains $k$ edge-disjoint spanning trees and another forest $F$ with $|E(F)|>\frac{δ-1}δ(|V(G)|-1)$, and if $F$ is not a spanning tree, then $F$ has a component with at least $δ$ edges.