Abstract
The Grothendieck constant
κ
(
G
)
of a graph
G
=
(
[
n
]
,
E
)
is the integrality gap of the canonical semidefinite relaxation of the integer program
max
x
∈
{
±
1
}
n
∑
i
j
∈
E
w
i
j
x
i
⋅
x
j
, replacing
±
1
variables by unit vectors. We show that
κ
(
G
)
=
g
/
(
g
−
2
)
cos
(
π
/
g
)
≤
3
/
2
when
G
has no
K
5
-minor and girth
g
; moreover,
κ
(
G
)
≤
κ
(
K
k
)
if the cut polytope of
G
is defined by inequalities supported by at most
k
points; lastly the worst case ratio of clique-web inequalities is bounded by 3.