Corrected bugs reported by Koubek.

diff --git a/TODO b/TODO
index a76d1b8e28c61ea1a57818f564f38fe72462db40..cb7c13b7f172055a661928174edd615dd36ed1f8 100644 (file)
--- a/TODO
+++ b/TODO
@@ -29,7 +29,7 @@ Patrice:
   Finding Minimum Spanning Trees. SIAM J. on Comp. 5(4) (1976) pp.
   724-742. who gave a linear time algorithm for planar graphs, extended by
   Tarjan in 1983 to proper minor closed classes (both quoted by Gustedt).
-  [XXX: Cannot get the paper.]
+  [XXX: The paper should be in the library at MS.]
 > In 3.1.12 and 3.1.16, you should make explicit the dependence of the
   running time  with respect, for instance, to the Hadwiger number of the
   graph or to the maximal density nabla(G) of a minor of the graph, as

diff --git a/adv.tex b/adv.tex
index a5a1e13bf0cae8e6fe63af3fa0eab5a0edeafee5..35d5d9b685a558f1bd8d9c627ccdff3c88d7981f 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -419,7 +419,7 @@ contract the edges of~this forest and iterate.
 \algin A~graph~$G$ with an edge comparison oracle.
 \:$T\=\emptyset$. \cmt{edges of the MST}
 \:$\ell(e)\=e$ for all edges~$e$. \cmt{edge labels as usually}
-\:$m_0\=m$.
+\:$m_0\=m$. \cmt{in the following, $n$ and $m$ will change with the graph}
 \:While $n>1$: \cmt{We will call iterations of this loop \df{phases}.}
 \::$F\=\emptyset$. \cmt{forest built in the current phase}
 \::$t\=2^{\lceil 2m_0/n \rceil}$. \cmt{the limit on heap size}
@@ -504,7 +504,7 @@ time (Lemma~\ref{ijphase}).
 The Iterated Jarn\'\i{}k's algorithm runs in time $\O(m\log^* n)$.
 
 \proof
-$\beta(m,n) \le \beta(1,n) \le \log^* n$.
+$\beta(m,n) \le \beta(n,n) \le \log^* n$.
 \qed
 
 \cor\id{ijdens}%

diff --git a/mst.tex b/mst.tex
index e0f4c46d3d5f8cd19205d607501d18426c96ef79..8a6a12a278b66ac23924bcbcf551e06a618ee5fb 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -328,8 +328,8 @@ and $V\setminus M$ contains no blue edges, therefore we can use the Blue rule.
 
 \nota\id{deltanota}%
 We will use $\delta(M)$ to denote the cut separating~$M$ from its complement.
-That is, $\delta(M) = E \cap (M \times (V\setminus M))$. We will also abbreviate
-$\delta(\{v\})$ as~$\delta(v)$.
+That is, $\delta(M) = \{ uv \in E \mid u\in M, v\not\in M \}$.
+We will also abbreviate $\delta(\{v\})$ as~$\delta(v)$.
 
 \thmn{Red-Blue correctness}%
 For any selection of rules, the Red-Blue procedure stops and the blue edges form

diff --git a/notation.tex b/notation.tex
index f8a512f4084c1fe92c33079deb8d72b33f1a10f5..e02cccf67e49f23f78e77d15a54e569d54adbcb5 100644 (file)
--- a/notation.tex
+++ b/notation.tex
@@ -113,7 +113,7 @@ better define it carefully, following \cite{diestel:gt}:
 
 \defn A~\df{multigraph} is an ordered triple $(V,E,M)$, where $V$~is the
 set of vertices, $E$~is the set of edges, taken as abstract objects disjoint
-with the vertices, and $M$ is a mapping $E\rightarrow V \cup {V \choose 2}$
+with the vertices, and $M$ is a mapping $E\rightarrow {V \choose 2} \cup {V \choose 1}$
 which assigns to each edge either a pair of vertices or a single vertex
 (if the edge is a loop).
 

diff --git a/ram.tex b/ram.tex
index 095d54d88e2216d26df9e885c20119bbc16d1e25..ac0a4acbb1a47936a0c459633abb3aad11bfccbe 100644 (file)
--- a/ram.tex
+++ b/ram.tex
@@ -266,7 +266,7 @@ set of edges in the current graph and then flatten the graph, all this in time $
 We have spared the technical details for this section, in which we are going to
 explain several rather general techniques based on bucket sorting.
 
-As we have already suggested in the proof of Lemma \ref{contbor}, contractions
+As we have already suggested in the proof of Lemma \ref{contiter}, contractions
 can be performed in linear time by building an~auxiliary graph and finding its
 connected components. We will thus take care only of the subsequent flattening.