行政院國家科學委員會專題研究計畫 成果報告
圖的直徑、連通性與邊數
計畫類別： 個別型計畫
計畫編號： NSC94-2115-M-216-001-
執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 中華大學企業管理學系
計畫主持人： 蔡明春 共同主持人： 裴文
計畫參與人員： 蔡明春, 裴文
報告類型： 精簡報告
處理方式： 本計畫可公開查詢
中 華 民 國 95 年 10 月 30 日
Diameter, Connectivity and Edges of Graphs
Ming-Chun Tsai
Abstract.
In the past years, we study edge number, maximum genus and decay number of graphs with given diameter and connectivity. We found that there exist some relations among edge number, connectivity and diameter. In
particular, we proved that q≥2p−11 G
3 ) (G =
for any diameter 3 graph with minimum degree δ , where , denote the edge number and vertex number of , respectively. This result and the past research intrigue us to study the edge number of graphs with given diameter and connectivity. In this project, we prove that if is a 3-regular
p q G
G (p,q)−
≤16
p ≤2p−8
graph of diameter 3, then and . We also show this bound is sharp. Finally, we conjecture that
q
8
2 −
≥ p
q for any diameter 3 (p,q)− 3
graph G with )
(G =
δ .
1. Preliminary
Throughout this project, a graph may have multiple edges or loops, but a multigraph contains multiple edges, but no loops, and a simple graphs contains neither multiple edges nor loops. Let the decay number of G, ς(G) (resp.
the Betti deficiency, ξ(G)) be the minimum number of components (resp.
odd size components) of a co-tree of a connected graph G. The invariant )
ξ(G was introduced to calculate the maximum genus γM(G) of G by the formula γM(G)=(β(G)−ξ(G)) 2, where β(G) is the Betti number. It is clear that ξ(G)≤ς(G) for any graph. In the past years, we study edge number, maximum genus and decay number with given diameter and connectivity. We find that there exist some relations among edge number, connectivity and diameter. This relations followed the study of Murty[4] in 1969. He proved the following theorem.
Theorem 1.1(Murty[4]) If G is a -connected, diameter 2 2
− ) ,
(p q graph, then
5
2 −
≥ p q
After the study of Murty[4], Bollobas and Harary【3】in 1976 defined that is the set of graphs with order , diameter at most and diameter at most if it is deleted any s vertices. Furthermore,
is defined by the minimum edge number of all graphs in )
, ' , , (p d d s
P p d
d′ )
, ' , , (p d d s f
) , ' , , (p d d s
P . Thus, f(p,d,d −1,n−1) denotes the minimum edge number of connected, diameter graphs. To simplify the notation, we use
to replace
−
n d
) , , (p d n
f f(p,d,p−1,n−1). By the definition, Theorem 1 show that f(p,2,2)≥2p−5. Next, Bollobas and Harary【3】obtained the following results.
Theorem 2 (Bollobas and Harary【3】)
, 2 )
1 )(
1 2( ) 1 , 2 ,
( 2
⎭⎬
⎫
⎩⎨
⎧ − + − +
≥ p n n n
n p
f
where {x} denotes the least integer not less than x.
Moreover, Bollobas[1] replaced -connected by minimum degree . He proved the following theorem.
n n
Theorem 3 (Bollobas [1])
Let G be a graph of order d and size g with minimum degree at least and diameter .
≥3
n d
) 1
( If 2≤d ≤2m for some m, then
) ( , ) 1
) 1 ( ( 1
2 C n m
n n n
q p m −
−
− + −
≥
where C(3,m)=4m+1and C(n,m)=nm+1for n≥4 )
2
( If 3≤d ≤2m+1,then
2
1 1
1
) 1 ( ) ) 1 ( ( 1
2
+ + +
=
+
−
− +
≥
∑
m m m
i
i
p n n
p n q
By the above theorem, 3
2
4 2 ) , ,
(p d n p p
f ≥ − .
In 2004, we study the 3-connected, diameter 3 graphs, and we obtained the following result.
Theorem 4 (Tsai and Fu [10])
If G is a diameter 3 (p,q)−graph with δ(G)=3, then 11
2 −
≥ p
q .
By the above theorem, we have f(p,3,3)≥2p−11. By Theorem 1 and
Theorem 4, we find that f(p,k,k)≥2p−c, where c is a constant for
. For , it is clear that . That is,
for some constant.
3 ,
=2
k k =4 f(p,k,k)≥2p
c p k k p
f( , , )≥2 −
On the other hand, Murty[4] found the extremal 2-connected, diameter 2 graphs with
− ) ,
(p q q≥2p−5.
Theorem 5 (Murty[4])
The following statements are equivalent for a graph G. (1) G is an extremal 2-connected graph of diameter 2
(2) is either the Petersen graph or is constructed by connecting all vertices of or to a new vertex by paths of length 2.
G
K2 K3
In [9], we found that Theorem 1 cloud be extended. We defined that a subset A of E(G) is E−minimal if any two different components of
are joined by at most one edge in . A
G− G
Theorem 6 (Tsai[9])
Let G be a 2-connected graph of diameter 2 and let A be an subset of . Then
minimal
E− E(G)
m(G/A)≥2n(G/A)−5+i(G/A),
where i(G/A) is the number of components in G−A containing at least two vertices.
By the above theorem, Fu, Tsai and Xuong[8] found the extremal 2-connected, diameter 2 graphs with ς(G)=4 and ξ(G)=4.
Theorem 7 (Fu, Tsai and Xuong[8])
Let G be a 2-connected graph of diameter 2. Then ς(G)=4 if and only if is an extremal 2-connected graph of diameter 2 with loops added to vertices.
G
Theorem 8 (Fu, Tsai and Xuong[8])
Let G be a 2-connected graph of diameter 2. Then ξ(G)=4 if and only if is an extremal 2-connected graph of diameter 2 at each vertex of which an odd number of loops added to vertices.
G
The past research intrigue us to study the edge number of graphs with given diameter and connectivity. Therefore, we have the following objectives in this project.
(1) Find the minimum edge number of graphs with given diameter and connectivity.
(2) Find the extremal graphs with given diameter and connectivity.
2. The Main Results
In order to improve Theorem 4, we consider 3-regular graphs of diameter, we get the following theorem.
Theorem 9. Let G be a 3-regular (p,q)−graph of diameter 3. Then
≤16 p .
Proof. For any u∈V(G), we denote Ni(u)={v∈V(G)|d(u,v)=i}, . Let be a vertex of and
3 , 2 ,
=1
i w G z be a vertex of with
minimum number of
)
3(w N )
( )
( 2
1 z N w
N ∩ . First if |N1(z)∩N2(w)|≥2, then for any vertex
3
| ) ( ) (
|N1 zi ∩N2 w = zi∈N3(w). Thus and
∈
∑
≤
≤
) (
2 1
3
12
| ) (
| 2
| ) (
|
w N z
i
i
w N z
N
∑
∈
≥
) (
3 1
3
| ) (
| 2
| ) (
|
w N z
i
i
w N z
N . This implies
that |N3(w)|≤6 and then p≤16. So it is sufficient to consider . We have following three cases..
1
| ) ( ) (
|N1 z ∩N2 w =
Case 1. . Then there are at most two vertices of which are adjacent to two vertex of . Thus
4
| ) (
|N2 z ≤ N2(w)
)
3(w
N |N3(w)|≤6 and then p≤16. Case 2. . Let be a vertex of which is adjacent to
two vertices of . We denote that 5
| ) (
|N2 z = y1 N2(w)
)
1(w
N N1(w)∩N1(y1)={x1,x2} and . Thus if
} { ) (
\ )
( 1 1 3
1 w N y x
N = z1∈N3(w)∩N1(y1), then there exists a
vertex of which is adjacent to one of .
We let this vertex be . Since ) ( )
( 1 1
3 w N z
N ∩ N2(z1)∩N1(y3)
z2 d(z2,xi)≤3 for i=1,2, the vertex in
) ( )
( 1 1 3
2 z N y
N ∩ must be adjacent to one of . This
implies that at most two vertices of which are adjacent to two vertices of . Thus
) ( )
( 1 1 2
1 x N x
N ∪
)
2(w N )
3(w
N |N3(w)|≤7 and then p≤16.
Case 3. |N2(z)|=6. We denote that N(w)={x1,x2,x3} and ,
} , , { )
(x1 w y1 y2
N = N(x2)={w,y3,y4}, N(x3)={w,y5,y6}, as in Figure 1. Without loss of generality, let z be adjacent to . Since
, the other two neighborhoods of y1
1
| ) ( ) (
|N1 z ∩N2 w = z are in . We
let them be and .
)
3(w N z1 z2
Figure 1.
z* z2 z1
z
y6
y5
y4
y3
y2
y1
x3
x2
x1
w
For any vertex , , is adjacent to one vertex of . If there exists one of neighborhoods of or in which is adjacent to one vertex of
zi i=1,2 zi yj N2(w)
z1 z2 N2(w)
z* N3(w)∩N3(z), let .
Assume that ,
) ( , 3 *
3
1y y z E G
z ∈
)
* (
z N
z ∉ i i=1,2 and is not adjacent to
. Since for any , is
adjacent to at least one vertex of , z*
} {
\ )
( 3
2 z y
N
zi∈ zi ∈N3(w)∩N1(z*)∩N1(yj) zi )
( }
{z ∪Nk z k =1,2, in order to keep , for all , must be adjacent to two vertices of
3 ) , (z* yj ≤
d j z*
} , , ,
{y1 y2 y5 y6 . Assume that is adjacent to a vertex and is adjacent to a vertex
z2 z3∈N3(w)\{z} z3 z4 ∈N3(w)\{z2}. Then in order to keep
for , is adjacent to neighborhoods of or both of and are adjacent to neighborhoods of . If is adjacent to neighborhoods of , but and are not adjacent to , then
3 ) , (z* zi ≤
d i=2,3,4 z3 z*
z2 z4 z* z3
z* z2 z4 y4
3 ) , (x2 z3 >
d . If is adjacent to neighborhoods of or is adjacent to , then for some
z3 z*, z2 z4
y4 d(zi,yj)>3 i=2 or 4, or 5,6. If is not adjacent to neighborhoods of , then both of and are adjacent to neighborhoods of . Thus This is a
contradiction. Hence there is no
2 ,
=1 j
z3 z* z2 z4
z* d(x2,z3)>3. } {
\ )
3(
3 N w z
z ∈ and z4 ∈N3(w)\{z2} such that is adjacent to a vertex and is adjacent to a vertex Now we can count .
z2 z3 z3 z4.
| ) (
|N3 w
| ) ( ) (
|
| ) ( ) (
|
| ) ( ) (
|
| } {
|
| ) (
|N3 w ≤ z + N1 z ∩N3 w + N2 z ∩N3 w + N3 z ∩N3 w
| ) ( ) ( ) (
|
| } , {
|
1+ z1 z2 + N1 z1 ∪N1 z2 ∩N3 w
≤
| ) ( ) ( ) ( } {
| z* ∪N2 z1 ∪N2 z2 ∩N3 w +
≤1+2+2+2=7.
otherwise, there exists no neighborhoods z2 in
For of or
which is adjacent to one vertex of
z1 N2(w) z* N3(w)∩N3(z). Thus
|N3(w)|≤|{z}|+|N1(z)∩N3(w)|+|N2(z)∩N3(w)|+|N3(z)∩N3(w)|
≤1+|{z1,z2}|+|N1(z1)∪N1(z2)∩N3(w)|
+|N2(z1)∪N2(z2)∩N3(w)| ≤1+2+2+2=7.
This implies . However, is a 3-regular graph, has no odd order. So we have
≤17
p G G
≤16 p .
Since the graph in Figure 2 is a 3-regure (16, 24)-graph of diameter, the bound in Theorem 9 is sharp.
Figure 2.
By Theorem 9, we have the following corollary.
Corollary 10. If G is a 3-regular (p,q)−graph of diameter 3, then 8
2 −
≤ p
q .
3-regular graphs of diameter 3 have minimum number of edges among diameter 3 graphs with δ(G)=3. So we have the following conjecture.
Conjecture. If G is a diameter 3 (p,q)−graph withδ(G)=3 , then 8
2 −
≥ p
q .
References
[1]B. Bollobas, Graphs with given diameter and minimal degree, Ars Combinatoria 2, 1976, 3-9.
[2]B. Bollobas, Extremal problems in graph theory, J. Graph Theory 1, 1977, 117-123.
[3] B. Bollobas,and F. Harary, Extremal problems with given diameter and connectivity, Ars Combinatoria 1, 1976, 281-296.
[4]U.S.R. Murty, On some extremal graphs, Acta Math. Acad. Sci. Hung. 19, 1969, 69-74.
[5]H.L. Fu, M. Skoviera and M.C. Tsai, The maximum genus, matchings and the cycle space of a graph, Czechoslovak Math. J. 48(123) 1998, 329-339.
[6]H.L. Fu, M.C. Tsai and N.H. Xuong, The decay number and the maximum genus of diameter 2
graphs, Discrete Math. 226, 2001, 191-197.
[7]H.L. Fu and M.C. Tsai, The maximum genus of a graph with given diameter and connectivity, Electronic Note in Discrete Math. Vol. 11, 2002.
[8]The decay number and the maximum genus of a graph, Math. Slovaca 42 (4), 1992, 392-406.
[9]M.C. Tsai, A study of maximum genus via diameter, Ph. D thesis, Chiao Tung University, 1996.
[10]M.C. Tsai and H.L. Fu, Edge number of 3-connected diameter 3 graphs, Proceedings of the Seventh International Symposium on Parallel Architectures, Algorithms and Networks.
