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research
The generalized 3-connectivity of Cartesian product graphs
Authors
Hengzhe Li
Xueliang Li
Yuefang Sun
Publication date
2 May 2011
Publisher
View
on
arXiv
Abstract
The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let
S
S
S
be a nonempty set of vertices of
G
G
G
, a collection
{
T
1
,
T
2
,
.
.
.
,
T
r
}
\{T_1,T_2,...,T_r\}
{
T
1
​
,
T
2
​
,
...
,
T
r
​
}
of trees in
G
G
G
is said to be internally disjoint trees connecting
S
S
S
if
E
(
T
i
)
∩
E
(
T
j
)
=
∅
E(T_i)\cap E(T_j)=\emptyset
E
(
T
i
​
)
∩
E
(
T
j
​
)
=
∅
and
V
(
T
i
)
∩
V
(
T
j
)
=
S
V(T_i)\cap V(T_j)=S
V
(
T
i
​
)
∩
V
(
T
j
​
)
=
S
for any pair of distinct integers
i
,
j
i,j
i
,
j
, where
1
≤
i
,
j
≤
r
1\leq i,j\leq r
1
≤
i
,
j
≤
r
. For an integer
k
k
k
with
2
≤
k
≤
n
2\leq k\leq n
2
≤
k
≤
n
, the
k
k
k
-connectivity
κ
k
(
G
)
\kappa_k(G)
κ
k
​
(
G
)
of
G
G
G
is the greatest positive integer
r
r
r
for which
G
G
G
contains at least
r
r
r
internally disjoint trees connecting
S
S
S
for any set
S
S
S
of
k
k
k
vertices of
G
G
G
. Obviously,
κ
2
(
G
)
=
κ
(
G
)
\kappa_2(G)=\kappa(G)
κ
2
​
(
G
)
=
κ
(
G
)
is the connectivity of
G
G
G
. Sabidussi showed that
κ
(
G
â–¡
H
)
≥
κ
(
G
)
+
κ
(
H
)
\kappa(G\Box H) \geq \kappa(G)+\kappa(H)
κ
(
G
â–¡
H
)
≥
κ
(
G
)
+
κ
(
H
)
for any two connected graphs
G
G
G
and
H
H
H
. In this paper, we first study the 3-connectivity of the Cartesian product of a graph
G
G
G
and a tree
T
T
T
, and show that
(
i
)
(i)
(
i
)
if
κ
3
(
G
)
=
κ
(
G
)
≥
1
\kappa_3(G)=\kappa(G)\geq 1
κ
3
​
(
G
)
=
κ
(
G
)
≥
1
, then
κ
3
(
G
â–¡
T
)
≥
κ
3
(
G
)
\kappa_3(G\Box T)\geq \kappa_3(G)
κ
3
​
(
G
â–¡
T
)
≥
κ
3
​
(
G
)
;
(
i
i
)
(ii)
(
ii
)
if
1
≤
κ
3
(
G
)
<
κ
(
G
)
1\leq \kappa_3(G)< \kappa(G)
1
≤
κ
3
​
(
G
)
<
κ
(
G
)
, then
κ
3
(
G
â–¡
T
)
≥
κ
3
(
G
)
+
1
\kappa_3(G\Box T)\geq \kappa_3(G)+1
κ
3
​
(
G
â–¡
T
)
≥
κ
3
​
(
G
)
+
1
. Furthermore, for any two connected graphs
G
G
G
and
H
H
H
with
κ
3
(
G
)
≥
κ
3
(
H
)
\kappa_3(G)\geq\kappa_3(H)
κ
3
​
(
G
)
≥
κ
3
​
(
H
)
, if
κ
(
G
)
>
κ
3
(
G
)
\kappa(G)>\kappa_3(G)
κ
(
G
)
>
κ
3
​
(
G
)
, then
κ
3
(
G
â–¡
H
)
≥
κ
3
(
G
)
+
κ
3
(
H
)
\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)
κ
3
​
(
G
â–¡
H
)
≥
κ
3
​
(
G
)
+
κ
3
​
(
H
)
; if
κ
(
G
)
=
κ
3
(
G
)
\kappa(G)=\kappa_3(G)
κ
(
G
)
=
κ
3
​
(
G
)
, then
κ
3
(
G
â–¡
H
)
≥
κ
3
(
G
)
+
κ
3
(
H
)
−
1
\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)-1
κ
3
​
(
G
â–¡
H
)
≥
κ
3
​
(
G
)
+
κ
3
​
(
H
)
−
1
. Our result could be seen as a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page
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