diff --git a/chapter13.tex b/chapter13.tex
index e87828b..f318a98 100644
--- a/chapter13.tex
+++ b/chapter13.tex
@@ -53,17 +53,17 @@ works in the following graph:
 \node[draw, circle] (2) at (4,3) {2};
 \node[draw, circle] (3) at (1,1) {3};
 \node[draw, circle] (4) at (4,1) {4};
-\node[draw, circle] (5) at (6,2) {5};
+\node[draw, circle] (5) at (6,2) {6};
 \node[color=red] at (1,3+0.55) {$0$};
 \node[color=red] at (4,3+0.55) {$\infty$};
 \node[color=red] at (1,1-0.55) {$\infty$};
 \node[color=red] at (4,1-0.55) {$\infty$};
 \node[color=red] at (6,2-0.55) {$\infty$};
-\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
+\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
 \path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
-\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
-\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
 \path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
 \path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
 \end{tikzpicture}
@@ -82,15 +82,15 @@ First, all edges from node 1 reduce distances:
 \node[draw, circle] (4) at (4,1) {4};
 \node[draw, circle] (5) at (6,2) {5};
 \node[color=red] at (1,3+0.55) {$0$};
-\node[color=red] at (4,3+0.55) {$2$};
+\node[color=red] at (4,3+0.55) {$5$};
 \node[color=red] at (1,1-0.55) {$3$};
 \node[color=red] at (4,1-0.55) {$7$};
 \node[color=red] at (6,2-0.55) {$\infty$};
-\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
+\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
 \path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
-\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
-\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
 \path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
 \path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
 
@@ -110,15 +110,15 @@ reduce distances:
 \node[draw, circle] (4) at (4,1) {4};
 \node[draw, circle] (5) at (6,2) {5};
 \node[color=red] at (1,3+0.55) {$0$};
-\node[color=red] at (4,3+0.55) {$2$};
+\node[color=red] at (4,3+0.55) {$5$};
 \node[color=red] at (1,1-0.55) {$3$};
-\node[color=red] at (4,1-0.55) {$1$};
+\node[color=red] at (4,1-0.55) {$4$};
 \node[color=red] at (6,2-0.55) {$7$};
-\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
+\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
 \path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
-\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
-\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
 \path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
 \path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
 
@@ -135,15 +135,15 @@ Finally, there is one more change:
 \node[draw, circle] (4) at (4,1) {4};
 \node[draw, circle] (5) at (6,2) {5};
 \node[color=red] at (1,3+0.55) {$0$};
-\node[color=red] at (4,3+0.55) {$2$};
+\node[color=red] at (4,3+0.55) {$5$};
 \node[color=red] at (1,1-0.55) {$3$};
-\node[color=red] at (4,1-0.55) {$1$};
-\node[color=red] at (6,2-0.55) {$3$};
-\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
+\node[color=red] at (4,1-0.55) {$4$};
+\node[color=red] at (6,2-0.55) {$6$};
+\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
 \path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
-\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
-\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
 \path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
 \path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
 
@@ -169,15 +169,15 @@ the following path:
 \node[draw, circle] (4) at (4,1) {4};
 \node[draw, circle] (5) at (6,2) {5};
 \node[color=red] at (1,3+0.55) {$0$};
-\node[color=red] at (4,3+0.55) {$2$};
+\node[color=red] at (4,3+0.55) {$5$};
 \node[color=red] at (1,1-0.55) {$3$};
-\node[color=red] at (4,1-0.55) {$1$};
-\node[color=red] at (6,2-0.55) {$3$};
-\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
+\node[color=red] at (4,1-0.55) {$4$};
+\node[color=red] at (6,2-0.55) {$6$};
+\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
 \path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
-\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
-\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
 \path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
 \path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);