Lau's Extension Theorem

Let $G$ be a loopless multigraph, with $S,R\subseteq V(G)$ such that $S\cap R=\emptyset$, $S$ is $Qk$-edge-connected, and every $r\in R$ has degree at least $Qk$, where $Q\geq 30$. Given a vertex $v$ and a subpartition $P_k(v)$ of $E(v)$ into $k$ parts, an $S$-subgraph extends $v$ when it satisfies the extension condition associated with the subpartition at $v$; a collection of subgraphs balances $S\cup R$ when their induced edge subpartitions are balanced at every vertex in $S\cup R$.

Lau's Extension Theorem. There are $k$ edge-disjoint $S$-subgraphs that extend $v$ and balance $S\cup R$ if either $v\in S$, $d(v)=Qk$, and $P_k(v)$ is a balanced edge subpartition of $E(v)$, or $N(v)\subseteq S\cup R$, $d(v)\leq Qk$, $v\notin S\cup R$, and there is no edge cut of size at most $Qk$ that breaks $G$ into $C_1,C_2$ with $S\subseteq C_2$, $v\in C_1$, and $R\cap C_1\neq\emptyset$.

The paper states that this theorem was used in Lau's proof of the $30k$ bound for Steiner forest packing, but identifies a mistake in it and provides a counterexample. Thus the asserted theorem is refuted as stated; the paper's corrected argument instead yields a $36k$ bound, improved to $35k$ when $k\geq 8$.