Abstract
AbstractA d-dimensional annulus graph with radii $$R_1$$
R
1
and $$R_2$$
R
2
(here $$R_2\ge R_1\ge 0$$
R
2
≥
R
1
≥
0
) is a graph embeddable in $$\mathbb R^d$$
R
d
so that two vertices u and v form an edge if and only if their images in the embedding are at distance in the interval $$[R_1, R_2]$$
[
R
1
,
R
2
]
. In this paper we show that the family $$\mathcal A_d(R_1,R_2)$$
A
d
(
R
1
,
R
2
)
of d-dimensional annulus graphs with radii $$R_1$$
R
1
and $$R_2$$
R
2
is uniquely characterised by $$R_2/R_1$$
R
2
/
R
1
when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of $$\mathcal A_d(R_1,R_2)$$
A
d
(
R
1
,
R
2
)
, we show that $$\sup _{G\in \mathcal A_d(R_1,R_2)} \chi (G)/\omega (G)$$
sup
G
∈
A
d
(
R
1
,
R
2
)
χ
(
G
)
/
ω
(
G
)
is given by $$\exp (O(d))$$
exp
(
O
(
d
)
)
for all $$R_1,R_2$$
R
1
,
R
2
satisfying $$R_2\ge R_1 > 0$$
R
2
≥
R
1
>
0
and also $$\exp (\Omega (d))$$
exp
(
Ω
(
d
)
)
if moreover $$R_2/R_1\ge 1.2$$
R
2
/
R
1
≥
1.2
.
Publisher
Springer Science and Business Media LLC
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