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Restricted Connectivity of Cartesian Product

Graphs

Laihuan Chen,

Jixiang Meng,

Yingzhi Tian,

Fengxia Liu

Abstract—For a connected graph G= (V(G), E(G)), a vertex set S ⊆ V(G) is a k-restricted vertex-cut if G−S is disconnected such that every componen-t of G−S has at least k vertices. The k-restricted connectivity κk(G) of the graph G is the

cardinal-ity of a minimum k-restricted vertex-cut of G. In this paper, we give the 3-restricted connectivity and the 4-restricted connectivity of the Cartesian product graphs, and we proposed two conjectures for general cases of thek-restricted connectivity of the Cartesian product graphs.

Keywords: restricted connectivity, Cartesian product, perfect matching.

1

Introduction

We follow [1] for graph-theoretical terminology and nota-tion not defined here. A network can be modelled by an undirected graphs with no loops or multiple edges. The connectivity is a classic measure of network reliability. In [4], Harary proposed conditional connectivity, which is a more refined index than the connectivity. In this paper,

we consider finite, undirected and simple graphs. LetG

be a graph with vertex setV(G) and edge setE(G), and

S be a non-empty subset ofV(G), thenS is avertex-cut

ifG−S is disconnected,S is ak-restricted vertex-cut if

G−S is disconnected and every component ofG−S has

at least kvertices, and S is a cyclic vertex-cut ifG−S

is disconnected and has at least two components contain-ing cycles. Theconnectivityκ(G) is defined as the mini-mum cardinality over all vertex-cuts ofG, thek-restricted connectivity κk(G) is defined as the minimum

cardinal-ity over all k-restricted vertex-cuts of G, and the cyclic connectivity κc(G) is defined as the minimum

cardinali-ty over all cyclic vertex-cuts of G. It should be pointed

out that not all connected graphs have the k-restricted

vertex-cut. A connected graphGis calledκk-connectedif

κk(G) exists. Thegirth g(G) of the graphGis the length

of its shortest cycle if Gcontains cycles. Results on the

restricted connectivity are referred to [5, 8–11, 13].

Foundation item: The research is supported by

NS-FC (No.11531011,11171283,11401510,11501487). Received: 12May2015. Address: College of Mathematics and

Sys-tem Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, P.R.China. TelFax: +86-0991-8582132. Corresponding author: mjxxju@sina.com.

Letu∈V(G) andG1,G2be two subgraphs ofG. Define N(u) ={v∈V(G)|vis adjacent tou}, andd(u) =|N(u)|

be the degree of u in G, and NG1(G2) = {v ∈ V(G1)\ V(G2)|vis adjacent to a vertex ofG2}. The graphGis

k-regularifd(u) =kfor anyu∈V(G). LetAbe a subset ofV(G). We defineG[A] as a subgraph ofGinduced by

A, N(A) ={u∈V(G)\A|uis adjacent to a vertex of

A}, andN[A] =A∪N(A).

Throughout this paper, we present the same notations related to the Cartesian product graphs as in [6]. As-sume G1 = (V1, E1) and G2 = (V2, E2), where V1 =

{x1, x2,· · · , xm} and V2 ={y1, y2,· · ·, yn}. The

Carte-sian product of graphs G1 and G2, denoted by G1G2,

is the graph with vertex set V1×V2 = {(x, y) | x∈ V1

and y ∈ V2} such that two vertices (x1, y1) and (x2, y2)

are adjacent if and only if eitherx1=x2withy1y2∈E2

ory1=y2withx1x2∈E1.

For convenience, we define two kinds of subgraphs G1y

andG2xofG1G2as follows: V(G1y) ={(x, y)|x∈V1}

and E(G1y) = {(x1, y)(x2, y) | x1x2 ∈ E1} for any

y ∈ V2, V(G2x) = {(x, y) | y ∈ V2} and E(G2x) = {(x, y1)(x, y2) | y1y2 ∈ E2} for any x ∈ V1.

Obvious-ly, G1y is isomorphic to G1 for any y ∈ V2, and G2x is

isomorphic toG2for anyx∈V1. By definition,V(G1y)∩

V(G1y0) =∅for anyy6=y0,V(G2x)∩V(G2x0) =∅for any x6=x0, V(G1y)∩V(G2x) = {(x, y)} for anyx∈V1 and y ∈ V2, and V(G1G2) = ∪y∈V2V1y = ∪x∈V1V2x. For

some results on the connectedness of Cartesian product graphs, see [2, 3, 6–8].

By the definition of the Cartesian product, the graph

G = G1G2 can be viewed as formed from m disjoint

copies ofG2, denoted byG2x1, G2x2, · · ·,G2xm, respec-tively, by connecting vertex (x, yi) of G2x with vertex

(x0, yi) ofG2x0 for anyyi∈V2wheneverxx0∈E1. These

new edges are called cross edges. That is, there exists

a perfect matching between two copiesG2x andG2x0 for

anyxx0 ∈E1. Similarly,Gcan also be viewed as formed

fromndisjoint copies ofG1, denoted byG1y1,G1y2,· · ·, G1yn, respectively, by connecting vertex (xi, y) of G1y with vertex (xi, y0) of G1y0 for any xi ∈ V1 whenever

yy0∈E2. Thus there is also a perfect matching between

two copiesG1y andG1y0 for anyyy0∈E2.

For S ⊆V(G1G2), letG01y =G1y−S for any y ∈V2,

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and G02x = G2x−S for any x ∈ V1. It is clear that

V(G01y) =V(G1y)\S for anyy∈V2,V(G02x) =V(G2x)\

S for anyx∈V1, and V(G1G2−S) =∪y∈V2V(G 0 1y) = ∪x∈V1V(G

0 2x).

In the graph theory, Menger’s theorem and Whitney Cri-terion are well-known [12]:

Theorem 1. (Menger’s theorem) Let G = (V, E) be a connected graph withx, y∈V. Then the minimum num-ber of vertices separating vertex x from vertex y in G

is equal to the maximum number of internally disjoint

(x, y)-paths inGif xy /∈E.

Theorem 2. (Whitney Criterion) If G is a graph with order at least k+ 1(k ≥ 1), then κ(G)≥ k if and only if there are at leastkinternally disjoint(x, y)-paths inG

for any x, y∈V.

2

Main Results

In this section, we always assume that G1 = (V1, E1)

and G2 = (V2, E2) have m and n vertices, respectively. For the 2-restricted connectivity of the Cartesian product graphs, we can find the following theorems in [6] and [8].

Theorem 3. ([6])κ2(KmKn)= min{m+ 2n−4,2m+

n−4} form+n≥6.

Theorem 4. ([8]) LetG2be ak2(≥2)-regular and

maxi-mally connected graph. Ifg(G2)≥4, thenκ2(KmG2) =

2k2+m−2 form≥2.

Theorem 5. ([8]) LetGi be aki(≥2)-regular and

max-imally connected graph with g(Gi) ≥ 4 for i ∈ {1,2}.

Thenκ2(G1G2) = 2k1+ 2k2−2.

In the following, we consider thek-restricted connectivity

of the Cartesian product graphs for k≥3.

By the exercise 2.1.9 in [1], we have|V(G)| ≥k2+ 1 for

g(G) ≥5, where k is the regularity of G, thus we have

the following lemma.

Lemma 2.1. Let G be a k-regular graph with girth

g(G)≥5. Then|V(G)| ≥3k−1fork≥2.

Theorem 6. (i) κ3(KmKn) =min{3m+n−6, m+

3n−6} form+n≥8 with m≥n+ 2 orn≥m+ 2;

(ii) κ3(KmKn) = 2m+ 2n−8 for m+n ≥ 8 with

m=n, orn=m+ 1, orm=n+ 1.

Proof. DenoteG=KmKn. AssumeG[A] is a

connect-ed subgraph ofGsuch that|A| ≥3.

(i)IfAis contained in one copyG2xorG1yforx∈V1and

y ∈ V2, then since |A| ≥3, G2x =∼Kn, and G1y ∼=Km,

there exist cycles in G[A]. Without loss of generality,

assume A ⊆ G2x. When |A| = 3, |N(A)| is minimum,

there are at least three vertices in each component of

G−N(A), and there exist cycles inG\N[A]. ThusN(A) is a 3-restricted vertex-cut, and also is a cyclic vertex-cut. Hence,κ3(G) =κc(G) =min{3m+n−6, m+ 3n−6}by

the theorem 2.2(i) in [2].

(ii)IfA is not contained in one copyG2x orG1y for any

x ∈ V1 and y ∈ V2. When there is no cycles in G[A],

G[A] = G[A1]∪G[A2] is a path of length k1 +k2 for

G2x ∼= Kn and G1y ∼= Km, where |Ai| = ki ≥ 2 for

i∈ {1,2}. AssumeG[A1]⊆G2x,G[A2]⊆G1y forx∈V1

and y ∈ V2, then |N(A)| = (m−1)k1 + (n−1)k2 −

k1k2+ 1. When k1 =k2 = 2, |N(A)| is minimum, and

|G\N[A]|>3. LetB = (G2x1∪G2x2)∩(G1y1∪G1y2) for x1, x2∈V1and y1, y2∈V2, then |N(B)|= 2m+ 2n−8,

|G\N[B]| >3, and there are cycles in G\N[B], thus

N(B) is a 3-restricted vertex-cut, and|N(A)|>|N(B)|,

a contradiction. We have A = B, and N(A) is a

3-restricted vertex-cut, and also is a cyclic vertex-cut of

G. Thusκ3(G) =κc(G) = 2m+ 2n−8 by the theorem

2.2(ii) in [2].

In the following, we consider the graphG=G1G2with

G1Km or G2Kn. ForG=KmG2, whenm= 2,

we easily obtainκ3(K2G2) = 3k2−1 fork2≥2, where

G2 is a k2-regular and maximally connected graph with

g(G2)≥5.

Lemma 2.2. Let G2 be a k2(≥ 2)-regular connected graph. Ifg(G2)≥5, then κ3(KmG2)≤3k2+m−3 for

m≥3.

Proof. Denote G=KmG2 and G1 =Km. The graph

G can be viewed as formed from m disjoint copies of

G2, denoted byG2x1, G2x2, · · ·, G2xm, respectively, by connecting vertex (x, yj) of G2x with vertex (x0, yj) of

G2x0 for any yj∈V2 andx6=x0.

When m = 3. Let P = u1u2u3 be a path of G, where

u1 = (x1, y1), u2 = (x1, y2), and u3 = (x2, y2). Since G2xi is a k2-regular graph fori∈ {1,2}, G1yj ∼=Kmfor

j∈ {1,2}, andg(G2)≥5, we have

|N({u1, u2, u3})| = |NG2x1({u1, u2})|+|NG2x2(u3)|

+ |NG1y1(u1)|+|NG1y2({u2, u3})|

− 1

= 2k2−2 +k2+m−2 +m−1−1

= 3k2+m−3.

Since g(G2) ≥ 5 and G1 ∼= Km, G−N[{u1, u2, u3}]

is connected, and |V(G)\N[{u1, u2, u3}]| ≥ 3. Hence

N({u1, u2, u3}) is a 3-restricted vertex-cut.

When m > 3. Let P = u1u2u3 be a path of G1y1 for y1 ∈ V2. Suppose ui = (xi, y1) for i ∈ {1,2,3}. Since

G1y1 ∼=Km, G2xi is a k2-regular graph for i∈ {1,2,3},

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andg(G2)≥5, we have

|N({u1, u2, u3})| = |NG1y1({u1, u2, u3})|

+ Σ3i=1|NG2xi({ui})| = 3k2+m−3.

Let G02x = G2x− {(x, y)} for some x ∈ {x1, x2, x3}

and y ∈ V2 \ {y1}. Since there is a perfect

match-ing between G2x and G2x0 for any x 6= x0, each vertex v ∈ V(G)\N[{u1, u2, u3}] is adjacent to some vertex in G02x, that is, G−N[{u1, u2, u3}] is connected, and

|V(G)\N[{u1, u2, u3}]| ≥3. Hence N({u1, u2, u3}) is a 3-restricted vertex-cut.

In conclusion,

κ3(G)≤ |N({u1, u2, u3})|= 3k2+m−3.

Theorem 7. LetG2 be ak2(≥2)-regular and maximally connected graph. Ifg(G2)≥5, thenκ3(KmG2) = 3k2+

m−3form≥3.

Proof. Denote G = KmG2 and G1 = Km. By

lem-ma 2.2, we have κ3(G) ≤ 3k2 +m−3. By

contrac-tion, κ3(G) <3k2+m−3. Assume S is a 3-restricted

vertex-cut of G with |S| = κ3(G). Let Sx = S∩V2x

for any x ∈ V1. Since G1 = Km, there exists a

per-fect matching between two copies G2x and G2x0 for any

x6=x0. IfSx=∅for somex∈V1, then all the vertices of G02xare connected toG02x(=G2x) by cross edges for any

x∈V1\{x}, which impliesG−Sis connected,

contradict-ing that S is a 3-restricted vertex-cut of G. Therefore,

Sx6=∅for anyx∈V1.

Letr be the number of copies ofG2 in Gwhich are

dis-connected in G−S. If r ≥ 3, then since G2 is

maxi-mally connected and Sx 6= ∅ for any x ∈ V1, we have

|S|= Σx∈V1|Sx| ≥3k2+m−3. We supposer≤2 in the

following.

Case 1. r= 0.

Without loss of generality, assumeG02x

1 ⊆CandG

0 2x2 ⊆ C0 forx1, x2 ∈V1, where C and C0 are distinct

compo-nents ofG−S. Since there is a perfect matching between

G2x1 and G2x2, we have|Sx1| ≥ |V(G 0

2x2)| =n− |Sx2|,

that is|Sx1|+|Sx2| ≥n. SinceSx6=∅for anyx∈V1, we

have

|S| = |Sx1|+|Sx2|+ Σx∈V1\{x1,x2}|Sx|

≥ n+m−2

≥ 3k2+m−3

forg(G2)≥5, a contradiction. HenceG−Sis connected, a contradiction.

Case 2. r= 1.

AssumeG02x1 is disconnected inG−Sforx1∈V1. Then

G02xis connected inG−Sfor anyx∈V1\ {x1}. Suppose

G02x2 is contained in a component C of G−S for x2 ∈

V1\ {x1}. If there exists a copy G02x3 *C forx3∈V1\

{x1, x2}, then since there is a perfect matching between G2x2 andG2x3, we have|Sx2|+|Sx3| ≥n, and

|S| = |Sx2|+|Sx3|+ Σx∈V1\{x2,x3}|Sx|

≥ n+m−2

≥ 3k2+m−3

for g(G2)≥5, a contradiction. Hence G02x ⊆C for any

x∈V1\ {x1}.

Denote the components of G02x1 byH1, H2, · · ·, Hl(l ≥

2). When|V(Hi)| ≤2 for somei∈ {1,2,· · · , l}. SinceS

is a 3-restricted vertex-cut ofG, we haveHi⊆C. When

|V(Hi)| ≥ 3k2 for some i ∈ {1,2,· · ·, l} and Hi * C, VG2x(Hi)⊆Sxfor anyx∈V1\ {x1}, and we have

|S| = Σx∈V1|Sx|

≥ κ(G2) + (m−1)|V(Hi)| ≥ k2+m−1 +|V(Hi)| −1

> 3k2+m−3,

a contradiction. When 3 ≤ |V(Hi)| ≤ 3k2−1 for some

i ∈ {1,2,· · ·, l} and Hi * C, VG2x(Hi) ⊆ Sx for any

x∈V1\ {x1}, and

|Sx1| ≥ |NG2x1[{u1, u2, u3}]| − |V(Hi)|

= 3k2−1− |V(Hi)|

for g(G2) ≥5, where P =u1u2u3 is a path of Hi. We

have

|S| = Σx∈V1|Sx|

≥ 3k2−1− |V(Hi)|+ (m−1)|V(Hi)| ≥ 3k2−1− |V(Hi)|+m−1 +|V(Hi)| −1

= 3k2+m−3,

a contradiction. Hence Hi ⊆ C for any i ∈

{1,2,· · ·, l}(l ≥ 2), and G−S is connected, a contra-diction.

Case 3. r= 2.

AssumeG02x1 andG02x2 are disconnected inG−S. Then

G02xis connected inG−Sfor anyx∈V1\ {x1, x2}, with the similar manner as the case 2, G02x is contained in a

component C of G−S, and if there are at least three

vertices in component H of G02x1 or G 0

2x2, then H ⊆C.

If H ⊆ G0

2x1 ∪G 0

2x2 and |H| ≤ 2, then H ⊆ C by S

being a 3-restricted vertex-cut ofG. Thus if H *C for

H ⊆ G02x

1 ∪G

0

2x2, then H is one of graphs in figure 1.

Since|N(F1)|= 3k2+2m−6,|N(F2)|= 4k2+2m−8, and

|N(F3)|= 4k2+ 3m−10, we have|N(Fi)| ≥3k2+m−3

for any i∈ {1,2,3}, a contradiction. HenceH ⊆C, and

G−S is connected, a contradiction.

Since all possible cases lead to a contradiction, we have

κ3(G) = 3k2+m−3.

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q

q q

q

q q

q

q q

q q

(x1, y1)

(x2, y1)(x2, y2)

(x1, y1) (x1, y2)

(x2, y1)(x2, y2) (x2, y2)(x2, y3)

[image:4.595.62.296.69.157.2]

(x1, y1) (x1, y2)

Figure 1. The subgraphs are not contained

F1 F2 F3

inC ofG−S.

Lemma 2.3. LetGibe aki(≥2)-regular connected graph

with g(Gi)≥5 fori∈ {1,2}. Then κ3(G1G2)≤3k1+

3k2−5.

Proof. Denote G=G1G2. Let P =u1u2u3 be a path

ofG, whereu1= (x1, y1),u2= (x2, y1), andu3= (x2, y2) forx1, x2∈V1andy1, y2∈V2. SinceG1yj is ak1-regular graph for j ∈ {1,2} and G2xi is a k2-regular graph for

i∈ {1,2}, we have

|N({u1, u2, u3})| = |NG2x1({u1})|+|NG2x2({u2, u3})|

+ |NG1y1({u1, u2})|+|NG1y2({u3})|

− 1

= 2k1−2 +k1+k2+ 2k2−2−1

= 3k1+ 3k2−5

forg(G1)≥5 andg(G2)≥5.

If there exists a isolated vertex (x0, y1) in G1y1 −

NG1y1({u1, u2}), then G 0

2x0 = G2x0. If there

exist-s a iexist-solated vertex (x2, y0) in G2x2 −NG2x2({u2, u3}),

then G01y0 = G1y0. Thus there are at least three

ver-tices in each component of G−N({u1, u2, u3}), that is,

N({u1, u2, u3}) is a 3-restricted vertex-cut ofG. Hence

κ3(G)≤ |N({u1, u2, u3})|= 3k1+ 3k2−5.

Theorem 8. Let Gi be aki(≥2)-regular and maximally

connected graph with g(Gi) ≥ 5 for i ∈ {1,2}. Then

κ3(G1G2) = 3k1+ 3k2−5.

Proof. Denote G = G1G2. By lemma 2.3, we have κ3(G) ≤ 3k1 + 3k2 − 5. By contraction, κ3(G) <

3k1+ 3k2−5. Assume S is a 3-restricted vertex-cut of G with |S|=κ3(G). Let Sx =S∩V2x for anyx∈ V1,

we have Sx 6= ∅ for any x ∈ V1. Otherwise, assume

Sx1 = ∅ for some x1 ∈ V1, and G

0

2x1(= G2x1) is

con-tained in a component C of G−S, then G02x ⊆ C for

anyx∈NG1(x1). For anyx 0 / N

G1[x1], we denote these

components ofG02x0 byH1,H2,· · ·,Hl. Since there exist

at least κ(G1) internally disjoint paths between x1 and

x0inG1, there areκ(G1)|V(Hi)|internally disjoint paths

betweenG02x1 andHiinG−Sx1−Sx0 fori∈ {1,2,· · ·, l}.

If|V(Hi)| ≥3k2, then Hi⊆C, otherwise

|S| ≥ κ(G1)|V(Hi)| ≥ 3k1k2

≥ 3(k1+k2−1)

> 3k1+ 3k2−5,

a contradiction. If 3≤ |V(Hi)| ≤3k2−1, andHi *C, then sinceNG2x0(Hi)⊆Sx0 andg(G2x0)≥5, we have

|Sx0| ≥ |NG

2x0[{u1, u2, u3}]| − |V(Hi)|

= 3k2−1− |V(Hi)|,

whereP =u1u2u3 is a path ofHi. Hence

|S| ≥ κ(G1)|V(Hi)|+ 3k2−1− |V(Hi)|

= |V(Hi)|(k1−1) + 3k2−1

> 3k1+ 3k2−5,

a contradiction.

If |V(Hi)| ≤2 for some i∈ {1,2,· · · , l}, then since S is

a 3-restricted vertex-cut ofG, there exists a copyG2x2 of

G2such that some vertex ofHi is adjacent to a vertex of

a component H ofG02x2 forx2∈V1\ {x

0}, and|V(H

i∪

H)| ≥3. Whenx2∈NG1[x1] or|H| ≥3, we haveHi⊆C

using the similar above manner. In the following, we

assume x2 ∈/ NG1[x1] and |H| ≤ 2, then F = Hi∪H

is one of graphs in figure 1. If F * C, then |N(F1)| =

3k1+ 3k2−5, and|N(F2)| =|N(F3)| = 4k1+ 4k2−8,

we have |N(Fi)| ≥3k1+ 3k2−5 for any i∈ {1,2,3}, a

contradiction. Thus F ⊆C, and G−S is connected, a

contradiction. HenceSx6=∅for any x∈V1.

Letr be the number of copies ofG2 in Gwhich are

dis-connected inG−S. Ifr≥3, then

|S| = Σx∈V1|Sx|

≥ 3k2+m−3

> 3k1+ 3k2−5

forg(G1)≥5. We supposer≤2 in the following.

Case 1. r= 0.

Without loss of generality, assume G02x1 is contained in

one componentCofG−Sforx1∈V1. By the minimality

of S, there exists one vertex v ∈ N(u)∩V(C0) for any

u∈Sx1, whereC

0 is a component ofGS different from

C. Assumev∈G02x

2 forx2∈V1\ {x1}, thenG 0 2x2⊆C

0,

and there exists a perfect matching between the copies

G2x1 andG2x2. Thus|Sx1|+|Sx2| ≥n, and

|S| = |Sx1|+|Sx2|+ Σx∈V1\{x1,x2}|Sx|

≥ n+m−2

> 3k1+ 3k2−5

byg(Gi)≥5 fori∈ {1,2}, a contradiction.

Case 2. r= 1.

Assume G02x1 is disconnected in G−S forx1∈V1. G02x

is connected in G−S for any x ∈ V1\ {x1}. There is

at least one copyG2x2 ofG2such that|Sx2|< κ(G2) for

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x2∈V1\ {x1}. Otherwise

|S| ≥ mκ(G2)

≥ (3k1−1)k2

= 3(k1−1)k2+ 2k2

≥ 3(k1−1 +k2−1) + 2k2

> 3k1+ 3k2−5,

a contradiction. Let G02x

2 be contained in a component

C of G−S. By the minimality of S, there exists one

vertex v∈N(u)∩V(C0) for anyu∈Sx2, whereC 0 is a

component ofG−S different from C. Assume v∈G02x3

forx3∈V1\ {x2}. Ifx36=x1, thenG02x3 ⊆C0, and there is a perfect matching between the copiesG2x2 andG2x3.

Thus|Sx2|+|Sx3| ≥n, and

|S| = |Sx2|+|Sx3|+ Σx∈V1\{x2,x3}|Sx|

≥ n+m−2

> 3k1+ 3k2−5

byg(Gi)≥5 fori∈ {1,2}, a contradiction. Hencex3 =

x1.

Letv belong to a componentH ofG0

2x1. SinceG 0 2x2 ⊆C

and v ∈C0, we haveNG2x2(H)⊆Sx2. When |V(H)| ≥

3k2,

|S| = |Sx2|+ Σx∈V1\{x2}|Sx|

≥ 3k2+m−1

> 3k1+ 3k2−5

for g(G1) ≥ 5, a contradiction. When 3 ≤ |V(H)| ≤

3k2−1,

|NG2x1(H)| ≥ |NG2x1[{u1, u2, u3}]| − |V(H)|

≥ 3k2−1− |V(H)|,

whereP =u1u2u3is a path ofH. SinceNG2x1(H)⊆Sx1

andNG2x2(H)⊆Sx2, we have

|S| = |Sx1|+|Sx2|+ Σx∈V1\{x1,x2}|Sx|

≥ 3k2−1− |V(H)|+|Sx2|+m−2 > 3k1+ 3k2−5

forg(G1)≥5, a contradiction. When|V(H)| ≤2. Since

Sis a 3-restricted vertex-cut ofG,vis adjacent to another vertexwin one copyG02x4 ofG−S forx4∈V1\ {x1, x2}. If|V(G02x4)| ≤ |Sx2|+ 2, then

|S| = |Sx1|+|Sx2|+|Sx4|+ Σx∈V1\{x1,x2,x4}|Sx|

≥ κ(G2) +|Sx2|+n− |V(G 0

2x4)|+m−3

≥ 3k1+ 3k2−5

byk2≥2 andg(Gi)≥5 fori∈ {1,2}, a contradiction. If |V(G02x4)| ≥ |Sx2|+ 3, then since there are (|V(G

0 2x4)| −

|Sx2|)κ(G1) disjoint paths betweenG 0

2x2 andG 0

2x4 inG− Sx2−Sx4, we have

|S| ≥ |Sx2|+|Sx4|+ (|V(G 0

2x4)| − |Sx2|)κ(G1)

= |Sx2|+n− |V(G 0

2x4)|+ (|V(G 0 2x4)|

− |Sx2|)κ(G1)

= n+ (|V(G02x

4)| − |Sx2|)(k1−1) > 3k1+ 3k2−5,

a contradiction.

Case 3. r= 2.

Assume G02x

1 and G

0

2x2 are disconnected in G−S for

x1, x2 ∈ V1, then G02x is connected for any x ∈ V1 \

{x1, x2}. Let G02x3 be contained in a component C of

G−S for x3 ∈ V1\ {x1, x2}. By the minimality of S, there exists one vertexv∈N(u)∩V(C0) for anyu∈Sx3,

whereC0is a component ofG−Sdifferent fromC. Using

the similar manner as the case 2, if there are at least three vertices in a component H of G02x1 or G02x2, then

we obtain H ⊆ C. If H * C for H ⊆ G02x1 ∪G

0 2x2,

then H is one of graphs in figure 1. Since |N(F1)| =

3k1+ 3k2−5, and|N(F2)| =|N(F3)| = 4k1+ 4k2−8, we have |N(Fi)| ≥3k1+ 3k2−5 for any i∈ {1,2,3}, a

contradiction.

Since all possible cases lead to a contradiction, we have

κ3(G) = 3k1+ 3k2−5.

Using the similar above manner, we can obtain the similar results for the 4-restricted connectivity of the Cartesian product graphs.

Theorem 9. (i) κ4(KmKn) =min{4m+n−8, m+

4n−8} form+n≥10 withn≥2m orm≥2n;

(ii) κ4(KmKn) = 2m+ 2n−8 for4≤m≤2n−1 and

4≤n≤2m−1 .

Theorem 10. LetG2be ak2(≥2)-regular and maximal-ly connected graph. If g(G2) ≥ 6, then κ4(KmG2) =

4k2+m−4 form≥4.

Theorem 11. LetGibe aki(≥2)-regular and maximally

connected graph with g(Gi) ≥ 6 for i ∈ {1,2}. Then

κ4(G1G2) = 4k1+ 4k2−8.

Let Gi be a ki(≥ 2)-regular and maximally

connect-ed graph. If g(Gi) ≥ k+ 2(k ≥ 3), then using the

similar manner as the lemmas 2.2 and 2.3, we have

κk(KmG2) ≤ kk2+m−k for m ≥ k(k ≤ 10), and

κk(G1G2) ≤ kk1+kk2−3k+ 4(k ≤ 10). Hence, we

have the following conjectures.

Conjecture 1. Let G2 be a k2(≥ 2)-regular and max-imally connected graph. If g(G2) ≥ k + 2, then

κk(KmG2) =kk2+m−k form≥k(k≥2).

IAENG International Journal of Applied Mathematics, 46:1, IJAM_46_1_08

(6)

Conjecture 2. Let Gi be a ki(≥ 2)-regular and

maxi-mally connected graph with g(Gi)≥k+ 2 fori∈ {1,2}.

Thenκk(G1G2) =kk1+kk2−3k+ 4(k≥2).

References

[1] Bondy, J.A., Murty, U.S.R., Graph Theory,

Grad-uate Texts in Mathematics 244, Springer, Berlin, 2008.

[2] Chen, L.H., Meng, J.X., Tian, Y.Z., “Cyclic vertex connectivity of Cartesian product graphs”, Accepted by Chinese Annals of Mathematics, Series A.

[3] Chiue, W.S., Shieh, B.C., “On connectivity of the

Cartesian product of two graphs”,Appl. Math.

Com-put., vol. 102, pp. 129-137, 1999.

[4] Harary, F.,“Conditional connectivity”, Networks,

13(1983), pp.347-357.

[5] Liu, J., Zhang, X.D., “Cube-connected complete

graphs”, IAENG International Journal of Applied

Mathematics, vol. 44, no. 3, pp. 134-136, 2014. [6] L¨u, M., Wu, C., Chen, G.L., L¨u, C., “On super

con-nectivity of Cartesian product graphs”, Networks,

vol. 52, pp. 78-87, 2008.

[7] ˇSpacapan, S., “Connectivity of Cartesian product

of graphs”, Appl. Math. Lett., vol. 21, pp. 682-685,

2008.

[8] Tian, Y.Z., Meng, J.X., “Restricted connectivity for

some interconnection networks”, Graphs and

Com-binatorics, vol. 31, no. 5, pp.1727-1737, 2015. [9] Wang, S.Y., Lin, S.W., Li, C.F., “Sufficient

con-ditions for super k-restricted edge connectivity in

graphs of diameter 2”, Discrete Mathematics, vol.

309, pp. 908-919, 2009.

[10] Wang, S.Y., Zhang, L., Lin, S.W., “k-restricted edge

connectivity in (p+1)-clique-free graphs”, Discrete

Applied Mathematics, vol. 181, pp. 255-259, 2015. [11] Wang, S.Y., Zhao, N.N., “Degree conditions for

graphs to be maximally k-restricted edge

connect-ed and super k-restrictconnect-ed connect-edge connectconnect-ed”,Discrete

Applied Mathematics, vol. 184, pp. 258-263, 2015.

[12] Xu, J.M., “Theory and application of graphs”,

K-luwer, Dordrecht, 2003.

[13] Yang, W.H., Tian, Y.Z., Li, H.Z., “The minimum restricted edge-connected graph and the minimum

size of graphs with a given edge-degree”, Discrete

Applied Mathematics, vol. 167, pp. 304-309, 2014.

IAENG International Journal of Applied Mathematics, 46:1, IJAM_46_1_08

Figure

Figure 1. The subgraphs are not contained

References

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