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Section 11.5
311
8. (a) The sum of
all
scores is the number of arcs (by Proposition 11.2.2) and in a tournament, this
number is
(~)
=
~n(n
 1) because every two vertices are connected by a single arc.
(b) Any
t
vertices VI,
... ,
Vt
determine a digraph which is IC
t
with an arrow on each arc. The number
of arcs in this digraph is
m
=
~
t( t  1). So, by Proposition 11.2.2, the sums of the scores of the
vertices VI,
...
,Vt
just counting games between these players is
~t(t
 1). But the score of each
Vi
(in the entire tournament) is at least its score in the smaller digraph, so
E
Si
::::
~t(t
 1).
9. (a) Suppose
V
and
w
are vertices in a transitive tournament
T.
Either
vw
or
wv
must be an arc so,
without loss of generality, we assume
vw
is an arc.
If
x
is any vertex such that
wx
is an arc, so
also is
vx
an arc, by transitivity. Remembering that
vw
is an arc, it follows that
S
(
v)
::::
S
(
w)
+
1.
In particular,
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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