On the spanning fan-connectivity of graphs

(1)

Contents lists available atScienceDirect

Discrete Applied Mathematics

journal homepage:www.elsevier.com/locate/dam

On the spanning fan-connectivity of graphs

I

Cheng-Kuan Lin

a,∗

, Jimmy J.M. Tan

a

, D. Frank Hsu

b

, Lih-Hsing Hsu

c

a_{Department of Computer Science, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC} b_{Department of Computer and Information Science, Fordham University, New York, NY 10023, USA}

c_{Department of Computer Science and Information Engineering, Providence University, Taichung, 43301, Taiwan, ROC}

a r t i c l e i n f o

Article history:

Received 1 August 2007

Received in revised form 30 July 2008 Accepted 11 November 2008 Available online 3 January 2009

Keywords: Hamiltonian connected Hamiltonian Dirac Theorem Menger Theorem Ore Theorem Connectivity Spanning connectivity Spanning fan-connectivity Spanning pipeline-connectivity Graph container a b s t r a c t

Let G be a graph. The connectivity of G,κ(G), is the maximum integer k such that there exists a k-container between any two different vertices. A k-container of G between u and v, Ck(u, v), is a set of k-internally-disjoint paths between u andv. A spanning container is a

container that spans V(G). A graph G is k∗_{-connected if there exists a spanning k-container} between any two different vertices. The spanning connectivity of G,κ∗₍_G₎_{, is the maximum} integer k such that G isw∗_{-connected for 1}_≤_{w ≤}_{k if G is 1}∗_-connected.

Let x be a vertex in G and let U= {y1,y2, . . . ,yk}be a subset of V(G)where x is not in U.

A spanning k−(x,U)-fan, Fk(x,U), is a set of internally-disjoint paths{P1,P2, . . . ,Pk}such

that Piis a path connecting x to yifor 1≤i≤k and∪ki=1V(Pi) =V(G). A graph G is k∗ -fan-connected (or k∗_f-connected) if there exists a spanning Fk(x,U)-fan for every choice of x and U with|U| =k and x6∈U. The spanning fan-connectivity of a graph G,κ∗

f(G), is defined as

the largest integer k such that G isw∗

f-connected for 1≤w ≤k if G is 1

∗

f-connected.

In this paper, some relationship between κ(G), κ∗(G), and κ_f∗(G) are discussed. Moreover, some sufficient conditions for a graph to be k∗

f-connected are presented.

Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be k∗-pipeline-connected.

Published by Elsevier B.V.

1. Introduction

For graph definitions and notations, we follow [1]. A graph G

=

(

V

,

E

)

consists of a finite set V

(=(

V

(

G

)))

and a subset

E

(=(

E

(

G

)))

of

{

(

u

, v) |

u

6=

v

and

(

u

, v)

is an unordered pair of elements of V

}

. We say that V is the vertex set and E is the

edge set of G. We use n

(

G

)

to denote

|

V

(

G

)|

. Two vertices u and

v

are adjacent if

(

u

, v) ∈

E. A graph H is a subgraph of graph G if V

(

H

) ⊆

V

(

G

)

and E

(

H

) ⊆

E

(

G

)

. Let S be a subset of V

(

G

)

. The subgraph of G induced by S, denoted G

[

S

]

, is the graph with the vertex set S and the edge set

{

(

u

, v) | (

u

, v) ∈

E

(

G

)

and u

, v ∈

S

}

. We use G

−

S to denote the graph G

[

V

(

G

) −

S

]

. A vertex cut is a set S

⊆

V

(

G

)

such that G

−

S has more than one component. A graph is k-connected if every vertex cut has at least k vertices. The connectivity of G,

κ(

G

)

, is the minimum size of a vertex cut. In other words,

κ(

G

)

is the maximum k such that G is k-connected. A complete graph has no cut set. We adopt the convention that

κ(

Kn

) =

n

−

1 where Knis the complete graph with n vertices. A path is a sequence of vertices represented by

h

v0, v1, . . . , v

_k

i

with no repeated vertex and

(v

i

, v

i+1)is an edge of G for 0

≤

i

≤

k

−

1. We also write the path

h

v0, v1, . . . , v

k

i

as

h

v0, . . . , v

i

,

Q

, v

j

, . . . , v

k

i

where Q is a subpath from

v

ito

v

j. A hamiltonian path of a graph G is a path that contains all vertices of V

(

G

)

. A graph G is hamiltonian connected if there is a hamiltonian path between every two different vertices. A cycle is a path with at least three vertices

I_{This work was supported in part by the National Science Council of the Republic of China under Contract NSC 96-2221-E-009-137-MY3.}

∗_{Corresponding author. Fax: +886 3 5721490.}

E-mail addresses:cklin@cs.nctu.edu.tw(C.-K. Lin),jmtan@cs.nctu.edu.tw(J.J.M. Tan),hsu@trill.cis.fordham.edu(D.F. Hsu),lhhsu@pu.edu.tw

(L.-H. Hsu).

0166-218X/$ – see front matter. Published by Elsevier B.V.

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such that the first vertex is the same as the last vertex. A hamiltonian cycle of G is a cycle that traverse every vertex of G. A graph is hamiltonian if it has a hamiltonian cycle. Let P

= h

x1,x2, . . . ,xk

i

be a path of the graph G connecting x1and xk. We use P−1_{to denote the path}

_h

_x

k

,

xk−1, . . . ,x1

i

. We use V

(

P

)

to denote the set

{

x1,x2, . . . ,xk

}

and I

(

P

)

to denote the set

V

(

P

) − {

x1,xk

}

. Let P1and P2be two paths of a graph G. We say that P1and P2are internally-disjoint if I

(

P1) ∩I

(

P2) = ∅.

Let u and

v

be two vertices of a graph G. A k-container of G between u and

v

, Ck

(

u

, v)

, is a set of k-internally-disjoint paths between u and

v

[2]. It follows from the Menger Theorem [3] that there is a k-container between any two distinct vertices of G if and only if G is k-connected. A k-container Ck

(

u

, v) = {

P1,P2, . . . ,Pk

}

of G is a k∗-container if

∪

ki=1V

(

Pi

) =

V

(

G

)

. A graph G is k∗_{-connected if there exists a k}∗_{-container between any two distinct vertices. The spanning connectivity of G,}

κ

∗

₍

_G

₎

_{, is defined as the largest integer k such that G is}

_w

∗_{-connected for 1}

_≤

_{w ≤}

_{k if G is a 1}∗_{-connected graph. It is} obvious that a 1∗-connected graph is actually a hamiltonian connected graph and that a 2∗-connected graph is actually a hamiltonian graph. Moreover, any 1∗-connected graph except K1and K2is 2∗-connected. Thus, the concept of a k∗-connected

graph is a hybrid concept of connectivity and hamiltonicity. Recently, the spanning connectivity of graphs have been studied extensively [4–10].

There is a Menger type theorem similar to the spanning connectivity of a graph. Let x be a vertex in a graph G and let U

= {

y1,y2, . . . ,yt

}

be a subset of V

(

G

)

where x is not in U. A t-

(

x

,

U

)

-fan, Ft

(

x

,

U

)

, is a set of internally-disjoint paths

{

P1,P2, . . . ,Pt

}

such that Piis a path connecting x and yifor 1

≤

i

≤

t. It is proved by Dirac [11] that a graph G is k-connected if and only if it has at least k

+

1 vertices and there exists a t-

(

x

,

U

)

-fan for every choice of x and U with

|

U

| ≤

k and x

6∈

U. Similarly, we can introduce the concept of a spanning fan. A spanning k-

(

x

,

U

)

-fan is a k-

(

x

,

U

)

-fan

{

P1,P2, . . . ,Pk

}

such that

∪

k

i=1V

(

Pi

) =

V

(

G

)

. A graph G is k∗-fan-connected (also written as k∗f-connected) if there exists a spanning k-

(

x

,

U

)

-fan for every choice of x and U with

|

U

| =

k and x

6∈

U. The spanning fan-connectivity of a graph G,

κ

∗

f

(

G

)

, is defined as the largest integer k such that G is

w

_f∗-connected for 1

≤

w ≤

k if G is a 1∗_f-connected graph. In this paper, some relationship among

κ(

G

)

,

κ

∗

₍

_G

₎

_{, and}

_κ

∗

f

(

G

)

are discussed. Moreover, some sufficient conditions for a graph to be k ∗

f-connected are presented. There is another Menger type theorem similar to the spanning connectivity and spanning fan-connectivity of a graph. Let U

= {

x1,x2, . . .xt

}

and W

= {

y1,y2, . . . ,yt

}

be two t-subsets of V

(

G

)

. A

(

U

,

W

)

-pipeline is a set of internally-disjoint paths

{

P1,P2, . . . ,Pt

}

such that Piis a path connecting xito yπ(i)where

π

is a permutation of

{

1

,

2

, . . . ,

t

}

. It is known that a graph G is k-connected if and only if it has at least k

+

1 vertices and there exists a

(

U

,

W

)

-pipeline for every choice of U and W

with

|

U

| = |

W

| ≤

k and U

6=

W . Similarly, we can introduce the concept of spanning pipeline. A spanning

(

U

,

W

)

-pipeline

is a

(

U

,

W

)

-pipeline

{

P1,P2, . . . ,Pk

}

such that

∪

ik=1V

(

Pi

) =

V

(

G

)

. A graph G is k∗-pipeline-connected (or k∗p-connected) if there exists a spanning

(

U

,

W

)

-pipeline for every choice of U and W with

|

U

| = |

W

| ≤

k and U

6=

W . The spanning pipeline-connectivity of a graph G,

κ

_p∗

(

G

)

, is defined as the largest integer k such that G is

w

∗_p-connected for 1

≤

w ≤

k if G is a 1∗

p-connected graph.

In Section2, we establish some relationships among

κ(

G

)

,

κ

∗

₍

_G

₎

_{, and}

_κ

∗

f

(

G

)

. Section3gives sufficient conditions for a graph to be k∗

f-connected. In Section4, spanning pipeline-connectivity is included. Section5gives an example to illustrate the differences between

κ(

G

)

,

κ

∗

(

G

)

,

κ

_f∗

(

G

)

, and

κ

_p∗

(

G

)

.

2. Relationship among

κ(

G

)

,

κ

∗

₍

_G

₎

_{, and}

_κ

∗

f

(

G

)

Let u be a vertex of G and let H be a subgraph of G. The neighborhood of u with respect to H, denoted by N_H

(

u

)

, is

{

v ∈

V

(

H

) | (

u

, v) ∈

E

(

G

)}

. We use dH

(

u

)

to denote

|

NH

(

u

)|

. For any vertex u, the degree of u in G is dG

(

u

)

. The minimum

degree of G, written

δ(

G

)

, is min

{

dG

(

x

) |

x

∈

V

}

. Let u and

v

be any two non-adjacent vertices of G, we use G

+

(

u

, v)

to denote the graph obtained from G by adding the edge

(

u

, v)

.

Lemma 1. Every 1∗-connected graph is 1∗_f-connected. Moreover, every 1∗_f-connected graph that is not K2is 2∗f-connected. Thus,

κ

∗

f

(

G

) ≥

2 if G is a hamiltonian connected graph with at least three vertices.

Proof. Let G be a 1∗

-connected graph with at least three vertices and let x be any vertex of G. Assume that U

= {

y

}

with x

6=

y. Obviously, there exists a hamiltonian path P1joining x and y. Apparently,

{

P1

}

forms a spanning 1-

(

x

,

U

)

-fan. Thus,

G is 1∗_f-connected. Assume that U

= {

y1,y2

}

with x

6∈

U. Let Q be a hamiltonian path of G connecting y1and y2. We write

Q as

h

y1,Q1,x

,

Q2,y2

i

. We set P1as

h

x

,

Q1−1

,

y1

i

and P2as

h

x

,

Q2,y2

i

. Then

{

P1,P2

}

forms a spanning 2-

(

x

,

U

)

-fan. Thus, G is

2∗_f-connected and

κ

_f∗

(

G

) ≥

2.

Theorem 1.

κ

∗

f

(

G

) ≤ κ

∗

₍

_G

_{) ≤ κ(}

_G

₎

_{for any 1}∗

f-connected graph. Moreover,

κ

∗

f

(

G

) = κ

∗

₍

_G

_{) = κ(}

_G

_{) =}

_n

₍

_G

_{) −}

_{1 if and only if}

G is a complete graph.

Proof. Obviously,

κ

∗

(

G

) ≤ κ(

G

)

. Now, we prove that

κ

_f∗

(

G

) ≤ κ

∗

(

G

)

. Assume that

κ

_f∗

(

G

) =

k. Let x and y be any two vertices of G. We need to show that there is a k∗_{-container of G between x and y.}

Suppose that k

=

1. Since G is 1∗_f-connected, there is a spanning 1-

(

x

, {

y

}

)

-fan,

{

P1

}

, of G. Then

{

P1

}

forms a spanning

container of G between x and y.

Suppose that k

≥

2. Let U0

_{= {}

_y_1,_y_{2, . . . ,}_y

k−1

}

be a set of

(

k

−

1

)

neighbors of y not containing x. We set U

=

U0

∪ {

y

}

. By

(3)

{

(

yi

,

y

) |

yi

∈

U0

}

to obtain a k∗-container between x to y. Hence, G is k∗-connected. Therefore,

κ

f∗

(

G

) ≤ κ

∗

₍

G

)

for every 1∗

f-connected graph.

Suppose that G is not a complete graph. There exists a vertex cut S of size

κ(

G

)

. Let x and y be any two vertices in different connected components of G

−

S. Obviously, y is not in any

(

x

,

S

)

-fan of G. Thus,

κ

∗

f

(

G

) < κ(

G

)

.

3. Some sufficient conditions for a graph to be k_f∗-connected

Since the concept of spanning fan-connectivity is a generalization of hamiltonicity, we review some previews results concerning hamiltonian graphs and hamiltonian connected graphs.

Lemma 2 ([12]). Every graph G with at least three vertices and

δ(

G

) ≥

n(₂G)is 2∗_{-connected. Moreover, every graph G with at} least four vertices and

δ(

G

) ≥

n(₂G)

+

1 is 1∗-connected.

Lemma 3 ([13,14]). Let u and

v

be two non-adjacent vertices of G with dG

(

u

) +

dG

(v) ≥

n

(

G

)

. Then G is 2∗-connected if and only if G

+

(

u

, v)

is 2∗_{-connected. Moreover, suppose that d}

G

(

u

) +

dG

(v) ≥

n

(

G

) +

1, then G is 1∗-connected if and only if

G

+

(

u

, v)

is 1∗_-connected.

Lemma 4 ([15]). A graph G is 2∗-connected if dG

(

u

) +

dG

(v) ≥

n

(

G

)

for all non-adjacent vertices u and

v

. Moreover, a graph G is 1∗_{-connected if d}

G

(

u

) +

dG

(v) ≥

n

(

G

) +

1 for all non-adjacent vertices u and

v

.

For comparison, we list the preview results concerning spanning connectivity.

Lemma 5 ([8]).

κ

∗

₍

_G

_{) ≥}

₂

_δ(

_G

_{) −}

_n

₍

_G

_{) +}

_{2 if} n(G)

2

+

1

≤

δ(

G

) ≤

n

(

G

) −

2.

Lemma 6 ([9]). Let k be a positive integer. Suppose that u and

v

are two non-adjacent vertices of G with dG

(

u

)+

dG

(v) ≥

n

(

G

)+

k. Then

κ

∗

(

G

) ≥

k

+

2 if and only if

κ

∗

(

G

+

(

u

, v)) ≥

k

+

2.

Lemma 7 ([9]). Let k be a positive integer. Then

κ

∗

₍

_G

_{) ≥}

_k

₊

_{2 if d}

G

(

u

) +

dG

(v) ≥

n

(

G

) +

k for all non-adjacent vertices u

and

v

.

Note thatLemma 5(Lemmas 6and7, respectively) generalizes the result ofLemma 2, (Lemmas 3and4, respectively) in spanning connectivity. The following theorem on spanning fan-connectivity is analogous to that on spanning connectivity inLemma 6[9].

Lemma 8. Let u and

v

(

u

) +

dG

(v) ≥

n

(

G

) +

1, and let x and y be any two distinct vertices of G. Then G has a hamiltonian path joining x to y if and only if G

+

(

u

, v)

has a hamiltonian path joining x to y.

Proof. Since every path in G is a path in G

+

(

u

, v)

, there is a hamiltonian path of G

+

(

u

, v)

joining x to y if G has a hamiltonian path joining x to y.

Suppose that there is a hamiltonian path P of G

+

(

u

, v)

joining x to y. We need to show that there is a hamiltonian path of G between x and y. If

(

u

, v) 6∈

E

(

P

)

, then P is a hamiltonian path of G between x and y. Thus, we consider that

(

u

, v) ∈

E

(

P

)

. Without loss of generality, we write P as

h

z1,z2, . . . ,zi

,

zi+1, . . . ,zn(G)

i

where z1

=

x, zi

=

u, zi+1

=

v

,

and zn(G)

=

y. Since dG

(

u

) +

dG

(v) ≥

n

(

G

) +

1, there is an index k in

{

1

,

2

, . . . ,

n

(

G

)} − {

i

−

1

,

i

,

i

+

1

}

such that

(

zi

,

zk

) ∈

E

(

G

)

and

(

zi+1,zk+1) ∈E

(

G

)

. We set R

= h

z1,z2, . . . ,zk

,

zi

,

zi−1, . . . ,zk+1,zi+1,zi+2, . . . ,zn(G)

i

if 1

≤

k

≤

i

−

2 and R

= h

z1,z2, . . . ,zi

,

zk

,

zk−1, . . . ,zi+1,zk+1,zk+2, . . . ,zn(G)

i

if i

+

2

≤

k

≤

n

(

G

)

. Then R is a hamiltonian path of G between x and y.

Theorem 2. Assume that k is a positive integer. Let u and

v

(

u

) +

dG

(v) ≥

n

(

G

) +

k. Then

κ

∗ f

(

G

) ≥

k

+

1 if and only if

κ

∗ f

(

G

+

(

u

, v)) ≥

k

+

1. Proof. Obviously,

κ

∗ f

(

G

+

(

u

, v)) ≥

k

+

1 if

κ

∗ f

(

G

) ≥

k

+

1. Suppose that

κ

∗

f

(

G

+

(

u

, v)) ≥

k

+

1. Let x be any vertex of G

and U

= {

y1,y2, . . . ,yt

}

be any subset of V

(

G

)

such that x

6∈

U and t

≤

k

+

1. We need to find a spanning t-

(

x

,

U

)

-fan of G. Since G

+

(

u

, v)

is

(

k

+

1

)

∗_f-connected, there exists a spanning t-

(

x

,

U

)

-fan

{

P1,P2, . . . ,Pt

}

of G

+

(

u

, v)

with Pijoining x to yifor 1

≤

i

≤

t. Obviously,

{

P1,P2, . . . ,Pt

}

is a spanning t-

(

x

,

U

)

-fan of G if

(

u

, v)

is not in

∪

it=1E

(

Pi

)

. Thus, we consider

(

u

, v) ∈ ∪

t_i₌₁E

(

Pi

)

. ByLemma 3, we can find a spanning

(

x

,

U

)

-fan of G if t

=

1

,

2. Thus, we consider the case t

≥

3. Without

loss of generality, we may assume that

(

u

, v) ∈

P1. Therefore, we can write P1as

h

x

,

H1,u

, v,

H2,y1

i

. Let Pi0

= h

w

i

,

Pi0

,

yi

i

be the path obtained from Piby deleting x. Thus, we can write Pias

h

x

, w

i

,

Pi0

,

yi

i

for 1

≤

i

≤

t. Note that x

6=

w

iand Pi

= h

yi

i

if

w

i

=

yifor every 2

≤

i

≤

t. Case 1: d_P0 i

(

u

) +

dP 0 i

(v) ≥

n

(

P 0

i

) +

2 for some 2

≤

i

≤

t. Without loss of generality, we may assume that dP₂0

(

u

) +

dP0₂

(v) ≥

n

(

P_{2) +}0 _{2. Obviously, n}

₍

_P_{2) ≥}0 _{2. We write P}0

2

= h

w2

=

z1,z2, . . . ,zr

=

y2

i

. We claim that there exists an index j in

{

1

,

2

, . . . ,

r

−

1

}

such that

(

zj

, v) ∈

E

(

G

)

and

(

zj+1,u

) ∈

E

(

G

)

. Suppose that this is not the case. Then dP20

(

u

) +

dP20

(v) ≤

(4)

Fig. 1. Illustration of case 1.

Fig. 2. Illustration of case 2.1.

We set Q1

= h

x

, w2

=

z1,z2, . . . ,zj

, v,

H2,y1

i

, Q2

= h

x

,

H1,u

,

zj+1,zj+2, . . . ,zr

=

y2

i

, and Qi

=

Pifor 3

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning t-

(

x

,

U

)

-fan of G. SeeFig. 1for illustration. Case 2: dP0 i

(

u

) +

dP0i

(v) ≤

n

(

P 0 i

) +

1 for every 2

≤

i

≤

t. Case 2.1: d_P0 i

(

u

) +

dP 0 i

(v) <

n

(

P 0

i

) +

1 for some 2

≤

i

≤

t. Without loss of generality, we may assume that dP₂0

(

u

) +

dP₂0

(v) ≤

n

(

P₂₎0 _{. Thus,} dP1

(

u

) +

dP1

(v) =

dG

(

u

) +

dG

(v) −

t

X

i=2

(

d_P0 i

(

u

) +

dP 0 i

(v))

=

dG

(

u

) +

dG

(v) − (

dP₂0

(

u

) +

dP0₂

(v)) −

t

X

i=3

(

d_P0 i

(

u

) +

dP 0 i

(v))

≥

n

(

G

) +

k

−

n

(

P2) −0 t

X

i=3

(

n

(

P_i0

) +

1

)

=

n

(

P1) +k

−

(

t

−

2

)

≥

n

(

P1) +1

.

ByLemma 8, there is a hamiltonian path Q1of G

[

P1

]

joining x to y1. We set Qi

=

Pifor 2

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning t-

(

x

,

U

)

-fan of G. SeeFig. 2for illustration.

Case 2.2: dP0

i

(

u

) +

dPi0

(v) =

n

(

P

0

i

) +

1 for every 2

≤

i

≤

t. We have

dP1

(

u

) +

dP1

(v) =

dG

(

u

) +

dG

(v) −

t

X

i=2

(

d_P0 i

(

u

) +

dP 0 i

(v))

=

n

(

G

) +

k

−

t

X

i=2

(

n

(

P_i0

) +

1

)

=

n

(

P1) +k

−

(

t

−

1

)

≥

n

(

P1).

(5)

Let R

= h

y1,H₂−1

, v,

u

,

H₁−1

,

x

, w2,

P2,0 y2

i

. Then

dR

(

u

) +

dR

(v) =

dP1

(

u

) +

dP1

(v) +

dP₂0

(

u

) +

dP0₂

(v)

≥

n

(

P1) +n

(

P2) +0 1

=

n

(

R

) +

1

.

ByLemma 8, there is a hamiltonian path W of G

[

R

]

joining y1to y2. Thus, W can be written as

h

y1,W1,x

,

W2,y2

i

. We set

Q1

= h

x

,

W1−1

,

y1

i

, Q2

= h

x

,

W2,y2

i

, and Qi

=

Pifor 3

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning

(

x

,

U

)

-fan of G. We note thatTheorem 2analogizes the result ofLemma 3in spanning fan-connectivity. ByTheorem 2, we can obtain the following theorem.

Theorem 3. Let k be a positive integer. Then

κ

∗

f

(

G

) ≥

k

+

1 if G is not the complete graph and dG

(

u

) +

dG

(v) ≥

n

(

G

) +

k for all non-adjacent vertices u and

v

.

Proof. Let Ec

(

G

)

be the set

{

e

|

e

6∈

E

(

G

)}

. Without loss of generality, we write Ec

= {

e1,e2, . . . ,em

}

. We set H0

=

G and

Hibeing the graph with V

(

Hi

) =

V

(

Hi−1)and E

(

Hi

) =

E

(

Hi−1) ∪ {ei

}

for every 1

≤

i

≤

m. Since Hmis isomorphic to the complete graph with n

(

G

)

vertices,

κ

∗

f

(

Hm

) ≥

k

+

1. ByTheorem 2,

κ

f∗

(

G

) = κ

∗

f

(

H0) ≥k

+

1. Note thatTheorem 3is an analogous result ofLemma 4in spanning fan-connectivity.

Theorem 4.

κ

∗

f

(

G

) ≥

2

δ(

G

) −

n

(

G

) +

1 if n(G)

2

+

1

≤

δ(

G

)

.

Proof. Suppose thatn(₂G)

+

1

≤

δ(

G

)

. Obviously,

δ(

G

) ≤

n

(

G

) −

1 and n

(

G

) ≥

4. Suppose that n

(

G

) =

2m for some integer m

≥

2. Then

δ(

G

) =

m

+

k for some integer k with 1

≤

k

≤

m

−

1. Obviously, dG

(

u

) +

dG

(v) ≥

2

δ(

G

) =

2m

+

2k for all two distinct vertices u and

v

in G. ByTheorem 3,

κ

_f∗

(

G

) ≥

2k

+

1

=

2

δ(

G

) −

n

(

G

) +

1. Suppose that n

(

G

) =

2m

+

1 for some integer m

≥

2. Then

δ(

G

) =

m

+

k

+

1 for some integer k with 1

≤

k

≤

m

−

1, and dG

(

u

) +

dG

(v) ≥

2

δ(

G

) =

2m

+

2k

+

2 for all two distinct vertices u and

v

κ

∗

f

(

G

) ≥

2k

+

2

=

2

δ(

G

) −

n

(

G

) +

1 and the theorem follows.

Theorem 4analogizes the result ofLemma 2in spanning fan-connectivity. Moreover, when

δ(

G

) =

n

(

G

) −

2, we have the following corollary.

Corollary 1.

κ

_f∗

(

G

) =

n

(

G

) −

3 if

δ(

G

) =

n

(

G

) −

2 and n

(

G

) ≥

5.

Proof. ByLemma 5,

κ

∗

₍

_G

_{) ≥}

_n

₍

_G

₎₋

_{2. Since n}

₍

_G

₎₋

₂

_≤

_κ

∗

₍

_G

_{) ≤ κ(}

_G

_{) ≤ δ(}

_G

_{) =}

_n

₍

_G

₎₋

_2,

_κ(

_G

_{) =}

_n

₍

_G

₎₋

_{2. By}_{Theorem 4}_,

κ

∗ f

(

G

) ≥

n

(

G

) −

3. ByTheorem 1,

κ

∗ f

(

G

) < κ(

G

)

. Thus,

κ

∗ f

(

G

) =

n

(

G

) −

3. 4. Spanning pipeline-connectivity

Similar to some recent works on the spanning connectivity [4–10] and the spanning fan-connectivity, studied in Section2, we study spanning pipeline-connectivity in this section.Lemma 9andTheorem 5are analogous toLemma 1andTheorem 1

respectively.

Lemma 9. Every 1∗

-connected graph is 1∗_p-connected.

Theorem 5.

κ

∗

p

(

G

) ≤ κ

∗

f

(

G

) ≤ κ

∗

₍

_G

_{) ≤ κ(}

_G

₎

_{for any 1}∗

p-connected graph. Moreover,

κ

∗

p

(

G

) = κ

∗

f

(

G

) = κ

∗

₍

_G

_{) = κ(}

_G

₎

_{if and}

only if G is a complete graph.

Theorem 6. Assume that k is a positive integer. Let u and

v

be two non-adjacent vertices of G. Suppose that dG

(

u

) +

dG

(v) ≥

n

(

G

) +

k. Then

κ

_p∗

(

G

) ≥

k if and only if

κ

_p∗

(

G

+

(

u

, v)) ≥

k.

Proof. Obviously,

κ

∗ p

(

G

+

(

u

, v)) ≥

k if

κ

∗ p

(

G

) ≥

k. Suppose that

κ

∗ p

(

G

+

(

u

, v)) ≥

k. Let U

= {

x1,x2, . . . ,xt

}

and

W

= {

y1,y2, . . . ,yt

}

be any two subsets of G such that U

6=

W and t

≤

k. We need to find a spanning

(

U

,

W

)

-pipeline of G. Since G

+

(

u

, v)

is k∗_p-connected, there exists a spanning

(

U

,

W

)

-pipeline of G

+

(

u

, v)

. Let

{

P1,P2, . . . ,Pt

}

be a spanning

(

U

,

W

)

-pipeline with Pi joining xito yπ(i)for 1

≤

i

≤

t. Without loss of generality, we assume that

π(

i

) =

i. Obviously,

{

P1,P2, . . . ,Pt

}

is a spanning

(

U

,

W

)

-pipeline of G if

(

u

, v)

is not in P. Thus, we consider the case that

(

u

, v)

is in P. By

Lemma 3, we can find a spanning

(

U

,

W

)

-pipeline of G if t

=

1. Thus, we consider the case t

≥

2.

Case 1: U

∩

W

= ∅

. Without loss of generality, we may assume that

(

u

, v) ∈

P1. Thus, we can write P1as

h

x1,H1,u

, v,

H2,y1

i

.

(Note that H1

= h

x

i

if x

=

u, and H2

= h

y

i

if y

=

v

.) Let Pi0be the path obtained from Piby deleting x and yi. Thus, we can write Pias

h

xi

,

Pi0

,

yi

i

for 1

≤

i

≤

t.

Case 1.1: dP1

(

u

) +

dP1

(v) ≥

n

(

P1) +1. WithLemma 8, there is a hamiltonian path Q1of G

[

P1

]

joining x1to y1. We set Qi

=

Pi for 2

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning

(

U

,

W

)

-pipeline of G. SeeFig. 3for illustration.

(6)

Fig. 3. Illustration of Case 1.1.

Fig. 4. Illustration of Case 1.2. Suppose that dPi

(

u

) +

dPi

(v) ≤

n

(

Pi

) +

1 for every 2

≤

i

≤

t. Then

dG

(

u

) +

dG

(v) =

dP1

(

u

) +

dP1

(v) +

t

X

i=2

(

dPi

(

u

) +

dPi

(v))

≤

n

(

P1) + t

X

i=2

(

n

(

Pi

) +

1

)

=

n

(

G

) +

t

−

1

≤

n

(

G

) +

k

−

1

.

We obtain a contradiction. Thus, dPi

(

u

) +

dPi

(v) ≥

n

(

Pi

) +

2 for some 2

≤

i

≤

t. Without loss of generality, we assume that dP2

(

u

) +

dP2

(v) ≥

n

(

P2) +2. Obviously, n

(

P

0

2) ≥ 2. We write P0

2

= h

x2

=

z1,z2, . . . ,zr

=

y2

i

. We claim that there

exists an index j in

{

1

,

2

, . . . ,

r

−

1

}

such that

(

zj

, v) ∈

E

(

G

)

and

(

zj+1,u

) ∈

E

(

G

)

. Suppose this is not the case. Then d_P0

2

(

u

) +

dP20

(v) ≤

r

+

r

−

(

r

−

1

) =

r

+

1

=

n

(

P

0

2) +1. We get a contradiction.

We set Q1

= h

x2

=

z1,z2, . . . ,zj

, v,

H2,y1

i

, Q2

= h

x1,H1,u

,

zj+1,zj+2, . . . ,zr

=

y2

i

, and Qi

=

Pifor 3

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning

(

U

,

W

)

-pipeline of G. SeeFig. 4for illustration.

Case 2: U

∩

W

6= ∅

. Let

|

U

∩

W

| =

r. Without loss of generality, we assume that xi

=

yi for t

−

r

+

1

≤

i

≤

t. Let G0

₌

_G

_[

_V

₍

_G

_{) − (}

_U

_∩

_W

_)]

_{, U}0

₌

_U

₋

_{W , and W}0

₌

_W

₋

_{U. Obviously, d}

G0

(

u

) +

dG0

(v) ≥

dG

(

u

) +

dG

(v) −

2r

≥

n

(

G

) +

k

−

2r

=

n

(

G0

_{) +}

_k

₋

_r,

_|

_U0

_{| = |}

_W0

_{| =}

_t

₋

_r

_≤

_k

₋

_{r, and U}0

_∩

_W0

_{= ∅}

_{. By Case 1, there exists a spanning}

(

U0

,

W0

)

-pipeline

{

Q1,Q2, . . . ,Qt−r

}

of G0. We set Qi

= h

xi

i

for t

−

r

+

1

≤

i

≤

t. Then

{

Q1,Q2, . . . ,Qt

}

forms a spanning

(

U

,

W

)

-pipeline of G.

We note thatTheorems 6–8are analogous toLemmas 3,4and2in spanning pipeline-connectivity respectively. By

Theorem 6, we can obtain the following theorem.

Theorem 7. Let k be a positive integer. Then

κ

∗

p

(

G

) ≥

k if G is not the complete graph and dG

(

u

) +

dG

(v) ≥

n

(

G

) +

k for all non-adjacent vertices u and

v

.

Proof. Let Ec

₍

_G

₎

_{be the set}

_{

_e

_|

_e

_6∈

_E

₍

_G

_)}

_{. Without loss of generality, we write E}c

_{= {}

_e_1,_e_{2, . . . ,}_e

m

}

. We set H0

=

G and

Hibeing the graph with V

(

Hi

) =

V

(

Hi−1)and E

(

Hi

) =

E

(

Hi−1) ∪ {ei

}

for every 1

≤

i

≤

m. Since Hmis isomorphic to the complete graph with n

(

G

)

vertices,

κ

∗

p

(

Hm

) ≥

k. ByTheorem 6,

κ

p∗

(

G

) = κ

∗

p

(

H0) ≥k.

Theorem 8.

κ

_p∗

(

G

) ≥

2

δ(

G

) −

n

(

G

)

if n(₂G)

+

1

≤

δ(

G

)

.

Proof. Sincen(₂G)

+

1

≤

δ(

G

)

and

δ(

G

) ≤

n

(

G

) −

1, n

(

G

) ≥

4. Suppose that n

(

G

) =

2m for some integer m

≥

2. Then

δ(

G

) =

m

+

k for some integer k with 1

≤

k

≤

m

−

1. Thus, dG

(

u

) +

dG

(v) ≥

2

δ(

G

) =

2m

+

2k for all two distinct vertices u and

v

κ

_p∗

(

G

) ≥

2k

=

2

δ(

G

) −

n

(

G

)

. Suppose that n

(

G

) =

2m

+

1 for some integer m

≥

2. Then

(7)

δ(

G

) =

m

+

k

+

1 for some integer k with 1

≤

k

≤

m

−

1, and dG

(

u

) +

dG

(v) ≥

2

δ(

G

) =

2m

+

2k

+

2 for all two distinct vertices u and

v

κ

∗

f

(

G

) ≥

2k

+

1

=

2

δ(

G

) −

n

(

G

)

and the theorem follows.

Corollary 2.

κ

∗

p

(

G

) =

n

(

G

) −

4 if

δ(

G

) =

n

(

G

) −

2 and n

(

G

) ≥

5.

Proof. ByTheorem 8,

κ

∗

p

(

G

) ≥

n

(

G

)−

4. Let V

(

G

) = {

x1,x2, . . . ,xn(G)

}

. Without loss of generality, we assume that

(

x1,x2) 6∈ E

(

G

)

. We set U

= {

x3,x5,x6, . . . ,xn(G)

}

and W

= {

x4,x5,x6. . . . ,xn(G)

}

. Obviously, U

6=

W and

|

U

| = |

W

| =

n

(

G

) −

3.

Since there is no hamiltonian path of G

[

V

(

G

) − {

x5,x6, . . . ,xn(G)

}]

joining x3to x4, there is no spanning

(

U

,

W

)

-pipeline

of G. Thus,

κ

∗

p

(

G

) =

n

(

G

) −

4.

5. An example

We use the following example to illustrate that

κ(

G

)

,

κ

∗

(

G

)

,

κ

_f∗

(

G

)

, and

κ

_p∗

(

G

)

are really different concepts and, in general, have different values.

Example 1. Suppose that n is a positive integer with n

≥

2. Let H

(

n

)

be the complete 3-partite graph K2n,2n,n−1with vertex

partite sets V1

= {

x1,x2, . . . ,x2n

}

, V2

= {

y1,y2, . . . ,y2n

}

, and V3

= {

z1,z2, . . . ,zn−1

}

. Let G

(

n

)

be the graph obtained from

H

(

n

)

by adding the edge set

{

(

zi

,

zj

) |

1

≤

i

6=

j

<

n

}

. Thus, G

[

V3

]

is the complete graph Kn−1. Obviously, n

(

G

(

n

)) =

5n

−

1,

δ(

G

(

n

)) =

2n

+

(

n

−

1

) =

3n

−

1, and

κ(

G

(

n

)) = δ(

G

(

n

))

. In the following, we will show that

κ

∗

₍

_G

₍

_n

_{)) =}

_n

₊

_1,

_κ

∗

f

(

G

(

n

)) =

n, and

κ

∗

p

(

G

(

n

)) =

n

−

1.

ByLemma 5,

κ

∗

(

G

(

n

)) ≥

2

δ(

G

(

n

)) −

n

(

G

(

n

)) +

2

=

n

+

1. To show

κ

∗

(

G

(

n

)) =

n

+

1, we claim that there is no

(

n

+

2

)

∗-container of G

(

n

)

between x1and x2. Suppose this is not the case. Let

{

P1,P2, . . . ,Pn+2

}

be an

(

n

+

2

)

∗-container

of G

(

n

)

between x1and x2. Obviously,

|

V

(

Pi

) ∩ (

V1

− {

x1,x2

}

)| ≤ |

V

(

Pi

) ∩ (

V2

∪

V3)| −1 for 1

≤

i

≤

n

+

2. Thus,

P

n+2

i=1

|

V

(

Pi

) ∩ (

V1

− {

x1,x2

}

)| = (P

n

+2

i=1

|

(

V2

∪

V3) ∩V

(

Pi

)|) − (

n

+

2

)

. Therefore,

|

V1

− {

x

,

y

}| ≤ |

V2

∪

V3

| −

(

n

+

2

)

.

However,

|

V1

− {

x

,

y

}| =

2n

−

2 but

|

V2

∪

V3

| −

(

n

+

2

) =

2n

−

3. This leads to a contradiction.

ByTheorem 4,

κ

∗

f

(

G

(

n

)) ≥

2

δ(

G

(

n

)) −

n

(

G

(

n

)) +

1

=

n. To show

κ

∗

f

(

G

(

n

)) =

n, we claim that there is no spanning

(

x1,U

)

-fan of G

(

n

)

where U

= {

y1,y2, . . . ,yn+1

}

. Suppose this is not the case. Let

{

P1,P2, . . . ,Pn+1

}

be a spanning

(

x1,U

)

-fan of G

(

n

)

. Without loss of generality, we assume that Pi is a path joining x1 to yi for 1

≤

i

≤

n

+

1. Obviously,

|

(

V1

− {

x1

}

) ∩

V

(

Pi

)| ≤ |((

V2

∪

V3) − {yi

}

) ∩

V

(

Pi

)|

. Thus,

P

n+1

i=1

|

(

V1

− {

x1

}

) ∩

V

(

Pi

)| ≤ P

n

+1

i=1

|

((

V2

∪

V3) − {yi

}

) ∩

V

(

Pi

)|

. Therefore,

|

V1

− {

x1

}| ≤ |

(

V2

∪

V3)−U

|

. However,

|

V1

− {

x1

}| =

2n

−

1 but

|

(

V2

∪

V3)−U

| =

2n

−

2. We get a contradiction.

ByTheorem 8,

κ

∗

p

(

G

(

n

)) ≥

2

δ(

G

(

n

)) −

n

(

G

(

n

)) =

n

−

1. To prove

κ

∗

p

(

G

(

n

)) =

n

−

1, we claim that there is no spanning

(

U

,

W

)

-pipeline where U

= {

x1,x2, . . . ,xn

}

and W

= {

xn+1,xn+2, . . . ,x2n

}

. Suppose that there exists a spanning

(

U

,

W

)

-pipeline

{

P1,P2, . . . ,Pn

}

. Obviously,

|

V2

∩

V

(

Pi

)| −

1

≤ |

V3

∩

V

(

Pi

)|

for 1

≤

i

≤

n. Then

P

n i=1(|V2

∩

V

(

Pi

)| −

1

) ≤

P

n i=1

|

V3

∩

V

(

Pi

)|

. Therefore,

(P

ni=1

|

V2

∩

V

(

Pi

)|) −

n

≤

P

n i=1

|

V3

∩

V

(

Pi

)|

. However,

(P

ni=1

|

V2

∩

V

(

Pi

)|) −

n

= |

V2

| −

n

=

n but

P

n i=1

|

V3

∩

V

(

Pi

)| = |

V3

| =

n

−

1. We get a contradiction. References

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