On the extremal number of edges in hamiltonian connected graphs

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Applied Mathematics Letters 23 (2010) 26–29

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Applied Mathematics Letters

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

On the extremal number of edges in hamiltonian connected graphs

I

Tung-Yang Ho

a,∗

, Cheng-Kuan Lin

b

, Jimmy J.M. Tan

b

, D. Frank Hsu

c

, Lih-Hsing Hsu

d

a_{Department of Information Management, Ta Hwa Institute of Technology, Hsinchu, 30740, Taiwan, ROC} b_{Department of Computer Science, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC} c_{Department of Computer and Information Science, Fordham University, New York, NY 10023, USA}

d_{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 19 March 2009 Accepted 19 March 2009 Keywords: Hamiltonian connected

Edge-fault tolerant hamiltonian connected

a b s t r a c t

Assume that n andδare positive integers with 3≤δ <n. Let hc(n, δ)be the minimum number of edges required to guarantee an n-vertex graph G with minimum degreeδ(G) ≥ δ

to be hamiltonian connected. Any n-vertex graph G withδ(G) ≥ δis hamiltonian connected if |E(G)| ≥ hc(n, δ). We prove that hc(n, δ) = C(n −δ + 1,2) + δ2₋_{δ +} _{1 if} δ ≤ bn+3×(n mod 2) 6 c+1, hc(n, δ) =C(n−b n 2c+1,2)+b n 2c 2 −bn 2c+1 ifb n+3×(n mod 2) 6 c+1< δ ≤ bn 2c, and hc(n, δ) = d nδ 2eifδ > b n 2c.

In this paper, we use C

(

a

,

b

)

to denote the combination of ‘‘a’’ numbers taking ‘‘b’’ numbers at a time, where a

,

b are

posi-tive integers and a

≥

b. For the graph definitions and notations, we follow [1]. Let G

=

(

V

,

E

)

be a graph if V is a finite set and

E is a subset of

{

(

u

, v) | (

u

, v)

is an unordered pair of V

}

. We say that V is the vertex set and E is the edge set. Two vertices u and

v

are adjacent if

(

u

, v) ∈

E. The complete graph Knis the graph with n vertices such that any two distinct vertices are adjacent.

The degree of a vertex u in G, denoted by deg_G

(

u

)

, is the number of vertices adjacent to u. We use

δ(

G

)

to denote min

{

deg_G

(

u

) |

u

∈

V

(

G

)}

. A path of length m

−

1,

h

v

₀

, v

₁

, . . . , v

_m−1

i

, is an ordered list of distinct vertices such that

v

iand

v

i+1are adjacent

for 0

≤

i

≤

m

−

2. A cycle is a path with at least three vertices such that the first vertex is the same as the last one. A

hamil-tonian cycle of G is a cycle that traverses every vertex of G exactly once. A graph is hamilhamil-tonian if it has a hamilhamil-tonian cycle. A hamiltonian path is a path of length

|

V

(

G

)|−

1. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. It is easy to see that a hamiltonian connected graph with at least three vertices is hamiltonian. It is proved by Moon [2] that the degree of any vertex in a hamiltonian connected graph with at least four vertices is at least 3. Therefore, it is natural to consider the n-vertex graph G with n

≥

4 and

δ(

G

) ≥

3. Assume that n and

δ

are positive integers with 3

≤

δ <

n. Let hc

(

n

, δ)

be the minimum number of edges required to guarantee an n-vertex graph with minimum degree

δ(

G

) ≥ δ

to be hamiltonian connected. Any n-vertex graph G with

δ(

G

) ≥ δ

is hamiltonian connected if

|

E

(

G

)| ≥

hc

(

n

, δ)

. We will prove the following main theorem.

Theorem A. Assume that n and

δ

are positive integers with 3

≤

δ <

n. Then

hc

(

n

, δ) =











C

(

n

−

δ +

1

,

2

) + δ

2

−

δ +

1 if

δ ≤

n

+

3

×

(

n mod 2

)

6

+

1

,

C

n

−

j

n 2

k

+

1

,

2

+

j

n 2

k

2

−

j

n 2

k

+

1 if

n

+

3

×

(

n mod 2

)

6

+

1

< δ ≤

j

n 2

k ,

d

n

δ/

2

e

if

δ >

j

n 2

k .

I_{This work was supported in part by the National Science Council of the Republic of China under Contract NSC 98-2115-M-233-001.}

∗_{Corresponding author.}

E-mail address:hoho@thit.edu.tw(T.-Y. Ho).

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T.-Y. Ho et al. / Applied Mathematics Letters 23 (2010) 26–29 27 We will defer the proof ofTheorem Ato Section4. In Section2, we describe an application ofTheorem A, which is the original motivation of this paper. In particular, we establish the relationship between hc

(

n

,

g

)

and g-conditional edge-fault tolerant hamiltonian connectivity of the complete graph Kn. In Section3, we present some preliminary results. Section4

gives the proof ofTheorem A.

2. An application

A hamiltonian graph G is k edge-fault tolerant hamiltonian if G

−

F remains hamiltonian for every F

⊂

E

(

G

)

with

|

F

| ≤

k.

The edge-fault tolerant hamiltonicity,He

(

G

)

, is defined as the maximum integer k such that G is k edge-fault hamiltonian if G is hamiltonian and is undefined otherwise. It is proved by Ore [3] that any n-vertex graph with at least C

(

n

,

2

) − (

n

−

3

)

edges is hamiltonian. Moreover, there exists an n-vertex non-hamiltonian graph with C

(

n

,

2

) − (

n

−

2

)

edges. In other words,He

(

Kn

) =

n

−

3 for n

≥

3. In Latifi et al. [4], it is proved thatHe

(

Qn

) =

n

−

2 for n

≥

2 where Qnis the n-dimensional

hypercube. In Li et al. [5], it is proved thatHe

(

Sn

) =

n

−

3 for n

≥

3 where Snis the n-dimensional star graph.

Chan and Lee [6] began the study of the existence of a hamiltonian cycle in a graph such that each vertex is incident with at least a number of nonfaulty edges. In particular, they have obtained results on hypercubes. A graph G is g-conditional k

edge-fault tolerant hamiltonian if G

−

F is hamiltonian for every F

⊂

E

(

G

)

with

|

F

| ≤

k and

δ(

G

−

F

) ≥

g. The g-conditional edge-fault tolerant hamiltonicity,Heg

(

G

)

, is defined as the maximum integer k such that G is g-conditional k edge-fault tolerant

hamiltonian if G is hamiltonian and is undefined otherwise. Chan and Lee [6] proved thatHeg

(

Qn

) ≤

2g−1

(

n

−

g

) −

1 for n

>

g

≥

2 and the equality holds for g

=

2.

Recently, Fu [7] study the 2-conditional edge-fault tolerant hamiltonicity of the complete graph. In the paper by the authors, Ho et al. [8] extend Fu’s result by studying the g-conditional edge-fault tolerant hamiltonicity of the complete graph for g

≥

2.

Several results (Lick [9], Moon [2], and Ore [10]) have studied hamiltonian connected graphs and some good sufficient conditions for a graph to be hamiltonian connected. Fault tolerant hamiltonian connectivity is another important parameter for graphs as indicated in [11]. A graph G is k edge-fault tolerant hamiltonian connected if G

−

F remains hamiltonian connected

for any F

⊂

E

(

G

)

with

|

F

| ≤

k. The edge-fault tolerant hamiltonian connectivity of a graph G,H Ce

(

G

)

, is defined as the

maximum integer k such that G is k edge-fault tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Again, Ore [10] proved thatH Ce

(

Kn

) =

n

−

4 for n

≥

4.

Similarly, a graph G is g-conditional k edge-fault tolerant hamiltonian connected if G

−

F is hamiltonian connected for every F

⊂

E

(

G

)

with

|

F

| ≤

k and

δ(

G

−

F

) ≥

g. The g-conditional edge-fault tolerant hamiltonian connectivity,H Cge

(

G

)

, is defined

to be the maximum integer k such that G is g-conditional k edge-fault tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise.

With the inspiration of the work by Fu [7] in the study of 2-conditional edge-fault tolerant hamiltonicity of the complete graph, Ho et al. [12] begin the study on 3-conditional edge-fault tolerant hamiltonian connectivity of the complete graph. The following result was obtained in [12]:

Let n

≥

4 and F

⊂

E

(

Kn

)

with

δ(

Kn

−

F

) ≥

3. Then Kn

−

F is hamiltonian connected if

|

F

| ≤

2n

−

10 for n

6∈ {

4

,

5

,

8

,

10

}

,

|

F

| =

0 for n

=

4,

|

F

| ≤

2 for n

=

5, and

|

F

| ≤

2n

−

11 for n

∈ {

8

,

10

}

.

We restate this result using our terminology.

Theorem 1.H C3

e

(

Kn

) =

2n

−

10 for n

6∈ {

4

,

5

,

8

,

10

}

and n

≥

5,H Ce3

(

K4

) =

0,H C3e

(

K5

) =

2,H C3e

(

K8

) =

5, and H C3e

(

K10

) =

9.

Now, we extend the result in [12] and use our main resultTheorem Ato computeH Cge

(

Kn

)

for 3

≤

g

<

n. Theorem 2.H Cge

(

Kn

) =

C

(

n

,

2

) −

hc

(

n

,

g

)

for 3

≤

g

<

n.

Proof. Let F be any faulty edge set of Knwith

|

F

| ≤

C

(

n

,

2

) −

hc

(

n

,

g

)

such that

δ(

Kn

−

F

) ≥

g. Obviously,

|

E

(

Kn

−

F

)| ≥

hc

(

n

,

g

)

. ByTheorem A, Kn

−

F is hamiltonian connected. Thus,H Cge

(

Kn

) ≥

C

(

n

,

2

) −

hc

(

n

,

g

)

.

Now, we prove thatH Cge

(

Kn

) ≤

C

(

n

,

2

) −

hc

(

n

,

g

)

. Assume thatH Cge

(

Kn

) ≥

C

(

n

,

2

) −

hc

(

n

,

g

) +

1. Let G be any

graph with hc

(

n

,

g

) −

1 edges such that

δ(

G

) ≥

g. Let F

=

E

(

Kn

) \

E

(

G

)

. In other words, G

=

Kn

−

F . Obviously,

|

F

| =

C

(

n

,

2

) −

hc

(

n

,

g

) +

1. SinceH Ceg

(

Kn

) ≥

C

(

n

,

2

) −

hc

(

n

,

g

) +

1, G is hamiltonian connected. This contradicts to

the definition of hc

(

n

,

g

)

. Thus,H Cge

(

Kn

) ≤

C

(

n

,

2

) −

hc

(

n

,

g

)

.

Therefore,H Cge

(

Kn

) =

C

(

n

,

2

) −

hc

(

n

,

g

)

for 3

≤

g

<

n. 3. Preliminary results

The following theorem is proved by Ore [10].

Theorem 3 ([10]). Let G be an n-vertex graph with

δ(

G

) > b

n₂

c

. Then G is hamiltonian connected.

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28 T.-Y. Ho et al. / Applied Mathematics Letters 23 (2010) 26–29

a

b

Fig. 1. The graphs (a) H3,11and (b) H4,12.

Theorem 4 ([9]). Let G be an n-vertex graph. Assume that the degree diof G satisfy d1

≤

d2

≤

. . . ≤

dn. If dj−1

≤

j

≤

n

/

2

⇒

dn−j

≥

n

−

j

+

1, then G is hamiltonian connected.

To our knowledge, no one has ever discussed the sharpness of the above theorem. In the following, we give a logically equivalent theorem.

Theorem 5. Let G be an n-vertex graph. Assume that the degree diof G satisfy d1

≤

d2

≤

. . . ≤

dn. If G is non-hamiltonian connected, then there exist at least one integer 2

≤

m

≤

n

/

2 such that dm−1

≤

m

≤

n

/

2 and dn−m

≤

n

−

m.

To discuss the sharpness ofTheorem 5, we introduce the following family of graphs. Let G1

=

(

V1

,

E1

)

and G2

=

(

V2

,

E2

)

be two graphs. The union of G1and G2, written G1

+

G2, has edge set E1

∪

E2and vertex set V1

∪

V2with V1

∩

V2

= ∅

. The join of G1and G2, written G1

∨

G2, obtained from G1

+

G2by joining each vertex of G1to each vertex of G2.

The degree sequence of an n-vertex graph is the list of vertices degree, in nondecreasing order, as d1

≤

d2

≤

. . . ≤

dn. For

2

≤

m

≤

n

/

2, let Hm,ndenote the graph

( ¯

Km−1

+

Kn−2m+1

) ∨

Km. The graphs H3,11and H4,12are shown inFig. 1. Obviously,

the degree sequence of Hm,nis

(

m

,

m

, . . . ,

m

|

{z

}

m−1

,

n

−

m

,

n

−

m

, . . . ,

n

−

m

|

{z

}

n−2m+1

,

n

−

1

,

n

−

1

, . . . ,

n

−

1

|

{z

}

m

)

A sequence of real numbers

(

p1

,

p2

, . . . ,

pn

)

is said to be majorised by another sequence

(

q1

,

q2

, . . . ,

qn

)

if pi

≤

qifor

1

≤

i

≤

n. A graph G is degree-majorised by a graph H if

|

V

(

G

)| = |

V

(

H

)|

and the nondecreasing degree sequence of

G is majorised by that of H. For instance, the 5-cycle is degree-majorised by the complete bipartite graph K2,3because

(

2

,

2

,

2

,

2

,

2

)

is majorised by

(

2

,

2

,

2

,

3

,

3

)

.

Lemma 1. Let G

=

(

V

,

E

)

be a graph, X be a subset of V , and u,

v

be any two distinct vertices in X . Suppose that there exists a hamiltonian path between u and

v

. Then there are at most

|

X

| −

1 connected components of G

−

X .

Let S be the subset of V

(

Hm,n

)

corresponding to the vertex of Km. Since 2

≤

m

≤

n

/

2,

|

S

| ≥

2. Let u and

v

be any

two distinct vertices in S. Obviously, there are m connected components of Hm,n

−

S. ByLemma 1, Hm,ndoes not have a

hamiltonian path between u and

v

. Thus, Hm,n is not hamiltonian connected. In other words, the result inTheorem 5is

sharp.

So we have the following corollary.

Corollary 1. The graph Hm,nis not hamiltonian connected where n and m are integers with 2

≤

m

≤

n

/

2.

Thus, the following theorem is equivalent toTheorem 5.

Theorem 6. If G is an n-vertex non-hamiltonian connected graph, then G is degree-majorised by some Hm,nwith 2

≤

m

≤

n

/

2. Corollary 2. Let n

≥

6. Assume that G is an n-vertex non-hamiltonian connected graph. Then

δ(

G

) ≤ b

n₂

c

and

|

E

(

G

)| ≤

max

{|

E

(

H_δ(G),n

)|, |

E

(

Hbn₂c,n

)|}

.

Proof. Let G be any n-vertex non-hamiltonian connected graph. WithTheorem 3,

δ(

G

) ≤ b

n₂

c

. ByTheorem 6, G is degree-majorised by some Hm,n. Since

δ(

Hm,n

) =

m,

δ(

G

) ≤

m

≤ b

n₂

c

. Therefore

|

E

(

G

)| ≤

max

{|

E

(

Hm,n

)| | δ(

G

) ≤

m

≤ b

n₂

c}

. Since

|

E

(

Hm,n

)| =

1₂

(

m

(

m

−

1

) + (

n

−

2m

+

1

)(

n

−

m

) +

m

(

n

−

1

))

is a quadratics function with respect to m and the maximum

value of it occurs at the boundary m

=

δ(

G

)

or m

= b

n

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T.-Y. Ho et al. / Applied Mathematics Letters 23 (2010) 26–29 29 ByCorollary 2, we have the following corollary.

Corollary 3. Let G be an n-vertex graph with n

≥

6. If

|

E

(

G

)| ≥

max

{|

E

(

H_δ(G),n

)|, |

E

(

Hbn₂c_,n

)|} +

1, then G is hamiltonian connected.

Lemma 2. Let n and k be integers with n

≥

6 and 3

≤

k

≤ b

n

2

c

. Then

|

E

(

Hk,n

)| ≥ |

E

(

Hbn₂c,n

)|

if and only if 3

≤

k

≤

b

n+3×(n mod 2)

6

c +

1 or k

= b

n

2

c

.

Proof. We first prove the case that n is even. We claim that

|

E

(

Hk,n

)| ≥ |

E

(

H₂n,n

)|

if and only if 3

≤

k

≤ b

n6

c +

1 or k

=

n

2.

Suppose that

|

E

(

Hk,n

)| < |

E

(

Hn₂,n

)|

. Then

|

E

(

Hk,n

)| =

1₂

(

k

(

k

−

1

) + (

n

−

2k

+

1

)(

n

−

k

) +

k

(

n

−

1

)) < |

E

(

Hn₂,n

)| =

1 2

((

n 2

−

1

)(

n 2

)) + (

n 2

)(

n

−

1

) + (

n 2

)

. This implies 3k 2

₋

₍

_2n

₊

₃

₎

_k

₊

₍

1 4n 2

₊

3 2n

) <

0, which means

(

k

−

n 2

)(

3k

−

n 2

−

3

) <

0. Thus

|

E

(

Hk,n

)| < |

E

(

Hn 2,n

)|

if and only if n 6

+

1

<

k

<

n

2. Note that n and k are integers with n is even, n

≥

6, and 3

≤

k

≤

n 2. Therefore,

|

E

(

Hk,n

)| ≥ |

E

(

Hn 2,n

)|

if and only if 3

≤

k

≤ b

n 6

c +

1 or k

=

n 2.

For odd integer n, using the same method, we can prove that

|

E

(

Hk,n

)| < |

E

(

Hn−1

2 ,n

)|

if and only if

n+3

6

+

1

<

k

<

n−1 2 .

Given that n

≥

7, and 3

≤

k

≤

n−1

2 , then

|

E

(

Hk,n

)| ≥ |

E

(

Hn−21,n

)|

if and only if 3

≤

k

≤ b

n+3

6

c +

1 or k

=

n−1

2 . Therefore,

the result follows. 4. Proof ofTheorem A

By brute force, we can check that hc

(

4

,

3

) =

6, hc

(

5

,

3

) =

8, and hc

(

5

,

4

) =

10. Therefore, the theorem holds for

n

=

4

,

5. Next, we consider the cases that 3

≤

δ ≤ b

n

2

c

and n

≥

6.

Suppose that 3

≤

δ ≤ b

n+3×(n mod 2)

6

c +

1 or

δ = b

n

2

c

. By Lemma 2,

|

E

(

Hδ,n

)| ≥ |

E

(

Hbn₂c,n

)|

. Let G be any

n-vertex graph with

δ(

G

) ≥ δ

and

|

E

(

G

)| ≥ |

E

(

H_δ,n

)| +

1. ByCorollary 3, G is hamiltonian connected. We note that

|

E

(

H_δ,n

)| +

1

=

C

(

n

−

δ +

1

,

2

) + δ

2

−

δ +

1. Therefore, hc

(

n

, δ) ≤

C

(

n

−

δ +

1

,

2

) + δ

2

−

δ +

1. ByCorollary 1, Hδ,nis not

hamiltonian connected. Thus, hc

(

n

, δ) > |

E

(

H_δ,n

)| =

C

(

n

−

δ +

1

,

2

)+δ

2

−

δ

. Hence, hc

(

n

, δ) =

C

(

n

−

δ +

1

,

2

)+δ

2

−

δ +

1.

Suppose that

b

n+3×(n mod 2)

6

c +

1

< δ < b

n

2

c

. ByLemma 2,

|

E

(

Hδ,n

)| < |

E

(

Hbn2c,n

)|

. Let G be any n-vertex graph with

δ(

G

) ≥ δ

and

|

E

(

G

)| ≥ |

E

(

Hbn₂c_,n

)| +

1. ByCorollary 3, G is hamiltonian connected. We note that

|

E

(

Hbn₂c_,n

)| +

1

=

C

(

n

−

b

n 2

c+

1

,

2

)+b

n 2

c

2

_−b

n 2

c+

1. Therefore, hc

(

n

, δ) ≤

C

(

n

−b

n 2

c+

1

,

2

)+b

n 2

c

2

_−b

n

2

c+

1. ByCorollary 1, Hbn₂c,nis not hamiltonian

connected. Thus, hc

(

n

, δ) > |

E

(

Hbn₂c,n

)| =

C

(

n

−b

n2

c+

1

,

2

)+b

n 2

c

2

_−b

n 2

c

. Hence, hc

(

n

, δ) =

C

(

n

−b

n 2

c+

1

,

2

)+b

n 2

c

2

_−b

n 2

c+

1.

Finally, we consider the case that

δ > b

n₂

c

and n

≥

6. Let G be any graph with

δ(

G

) ≥ δ > b

n₂

c

. ByTheorem 3, G is hamiltonian connected. Obviously,

|

E

(

G

)| ≥ d

n₂δ

e

. Thus, hc

(

n

, δ) = d

n₂δ

e

.

The proof of our main result,Theorem A, is complete. References

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