The Higher-Order Edge Toughness of a Graph and Truncated Uniformly Dense Matroids

In [Discrete Math. 111 (1993), 113-123], the \(c\)-th order edge toughness of a graph \(G\) is defined as
\[
\tau_c(G) = \min_{\substack{X \subseteq E(G), \&\omega(G – X) > c }} \left\{\frac{|X|}{\omega(G – X) – c}\right\},
\]
for any \(1 \leq c \leq |V(G)| – 1\). It is proved that \(\tau_c(G) \geq k\) if and only if \(G\) has \(k\) edge-disjoint spanning forests with exactly \(c\) components, and that for a given graph \(G\) with \(s = |E(G)|/(|V(G)| – c)\) and \(1 \leq c \leq |E(G)|\),
\(\tau_c(G) = s\) if and only if \(|E(H)| \leq s(|V(H)| – 1)\) for any subgraph \(H\) of \(G\). In this note, we shall present short proofs of the abovementioned theorems and shall indicate that these results can be extended to matroids.