MATH 682
Notes
Combinatorics and Graph Theory II
1 Minimal Examples and Extremal Problems
Minimal and extremal problems are really variations on the same question: what is the largest or
smallest graph you can ﬁnd which either avoids or satisﬁes some graph property? “Size” in these
contexts is usually “number of vertices” but in some contexts is “number of edges” on a graph with
a predetermined number of vertices.
1.1 Minimal examples
If we are trying to ﬁnd the smallest graph satisfying a certain property, it’s generally called a
minimal example
. For most properties, the minimal examples are pretty dull: whether we are
minimizing on number of vertices or number of dges among a ﬁxed set of vertices, most of our
graph paremeters have simple minimal examples. The smallest graph of diameter
d
is a path on
d
+ 1 vertices; the smallest graph of chromatic number
k
or clique number
k
is a connected graph
on
k
vertices, and the smallest graph of independence number
k
is
k
nonadjacent vertices. These
are selfevident and not terribly interesting.
The one parameter we have seen so far with an interesting and not too diﬃcult minimal example
is the parameter of connectivity. Finding the fewest number of vertices on which we may connect
a graph of connectivity
k
is not too interesting: once again, a complete graph
K
k
+1
is best, but if
we ﬁx a number of vertices
n
≥
k
+ 1, we might ask how many edges a graph on
n
vertices must
have in order to have connectivity
k
? It is easy to ﬁnd a necessary number of edges.
Proposition 1.
If

G

=
n
and
κ
(
G
) =
k
, then
k
G
k ≥ d
nk
2
e
.
Proof.
It has been seen previously, when ﬁrst introducing the concept of connectivity, that
κ
(
G
)
≥
δ
(
G
), so
δ
(
G
)
≥
k
. Then
2
k
G
k
=
X
v
∈
V
(
G
)
d
(
v
)
≥
X
v
∈
V
(
G
)
δ
(
G
) =

G

δ
(
G
)
≥
nk
and thus
k
G
k ≥
nk
2
, and since
k
G
k
must be an integer, we may take the ceiling of this lower
bound.
This almosttrivial bound will in fact completely satisfy our purposes, because we can show that
speciﬁc graphs achieve it.
Deﬁnition 1.
The
Harary graph
H
n,k
is a graph on the
n
vertices
{
v
1
,v
2
,...,v
n
}
deﬁned by the
following construction:
•
If
k
is even, then each vertex
v
i
is adjacent to
v
i
±
1
,
v
i
±
2
,. . . ,
v
i
±
k
2
, where the indices are
subjected to the wraparound convention that
v
i
≡
v
i
+
n
(e.g.
v
n
+3
represents
v
3
).
•
If
k
is odd and
n
is even, then
H
n,k
is
H
n,k

1
with additional adjacencies between each
v
i
and
v
i
+
n
2
for each
i
.
•
If
k
and
n
are both odd, then
H
n,k
is
H
n,k

1
with additional adjacencies
{
v
1
,v
1+
n

1
2
}
,
{
v
1
,v
1+
n
+1
2
}
,
{
v
2
,v
2+
n
+1
2
}
,
{
v
3
,v
3+
n
+1
2
}
,
···
,
{
v
n

1
2
,v
n
}
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March 25, 2010