• No results found

Some panconnected and pancyclic properties of graphs with a local ore-type condition

N/A
N/A
Protected

Academic year: 2021

Share "Some panconnected and pancyclic properties of graphs with a local ore-type condition"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

 

 

Some panconnected and pancyclic properties of 

graphs with a local ore‐type condition 

Armen S. Asratian and G. V. Sarkisian

The self-archived postprint version of this journal article is available at Linköping

University Institutional Repository (DiVA):

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-143289

  

  

N.B.: When citing this work, cite the original publication.

Asratian, A. S., Sarkisian, G. V., (1996), Some panconnected and pancyclic properties of graphs with a local ore-type condition, Graphs and Combinatorics, 12(3), 209-219.

https://doi.org/10.1007/BF01858455

Original publication available at:

https://doi.org/10.1007/BF01858455

Copyright:

Publisher URL Missing

 

 

(2)

Some Panconnected and Pancyclic Properties of Graphs

with a Local Ore-type Condition

A.S. Asratian

l

.

2

and G.V. Sarkisian

2

1

Department of Mathematics, University ofUmea, S-901 87 Umea, Sweden

2

Department of Mathematical Cybernetics, Yerevan State University, Yerevan, 375049,

Republic of Armenia

Abstract.

Asratian and Khachatrian proved that a connected graph G of order at least 3 is

hamiltonian if

d(u)

+

d(v);;:::

JN(u) U

N(v)

U N.(w)I for any path

uwv

with

uv

ff

E(G), .where

N(x)

is the neighborhood of a vertex

x.

We prove that a graph

G

with this condition, which is not complete bipartite, has the

following properties:

a) For each pair of vertices

x, y

with distance

d(x, y)

2:: 3 and for each integer

n, d(x, y) :;;; n

:;;; I V(G)I - 1, there is an

x

-

y path of length n.

(b) For each edge

e

which does not lie on a triangle and for each

n,

4 :;;;

n

:;;; I V(G)I, there

is a cycle of length

n

containing

e.

(c) Each vertex oft;; lies on a cycle of every length from 4 to

I

V(G)j.

This implies that G is vertex pancyclic if and only if each vertex of

G

lies on a triangle.

1. Introduction

We use Bondy and Murty [6] for terminology and notation not defined here and

consider finite simple graphs only. For each vertex

u

of a graph G we denote by

N(u)

the set of all vertices of

G

adjacent to

u.

The distance between vertices

u

and

vis

denoted by

d(u,

v). A path with

x

and

y

as end vertices is called an

x

-

y

path.

An

x

-

y

path is called a Hamilton path if it contains all the vertices of G. A graph

G

is Hamilton-connected if every two vertices of

G

are connected by a Hamilton

path.

Let G be a graph of order

p

;;:::: 3. G is called panconnected if for each pair of

distinct vertices x and

y

and for each I, d(x, y) ::::; / ::::;

p

- 1, there is an x -

y

path

of length l in G. G is called pancyclic if it contains a cycle of length l for each l

satisfying 3 :s;

l

:;;;

p. G

is called vertex pancyclic (edge pancyclic) if each vertex

(edge) of G lies on a cycle of every length from 3

top

inclusive.

Let M

1

= {K

n

,

n

,n;;::: 2} and J1

2

= {G: K

n

,

n

s;;; G s;;; K

n V

K�,n;;::: 3}.

The following results are known.

Theorem 1 (Ore [10]).

Let G be a graph of order p

;;:::: 3,

where d(u)

+

d(v)

;;::::

p for each pair of nonadjacent vertices u and v. Then G is hamiltonian.

(3)

210 A.S. Asratian and G.V. Sarkisian A graph G satisfying the condition of Theorem 1 is called an Ore graph.

Theorem 2 (Bondy [5]). An Ore graph G is pancyclic if. and only if G ~ ~¢1.

Note that Theorem 2 is a corollary from more general theorems of Bondy [5] and H/iggkvist, Faudree and Schelp [8].

Theorem 3 [7, 11]. Let G be an Ore graph of order p >_ 4. Then each vertex of G lies

on a cycle of every length from 4 to p inclusive, unless G ~ vg[ 1.

Theorem 4 (Asratian I and Khachatrian [9]). Let G be a connected graph of order

at least 3 where d(u) + d(v) >_ [N(u) U N(v) U N(w)[ for any path uwv with uv ¢ E(G). Then G is hamiltonian.

A simpler proof of Theorem 4 was suggested in [2]. Clearly, Theorem 4 implies Theorem 1. Moreover, while Theorem 1 only applies to graphs G with diameter 2 and large edge density (IE(G)I > ¼" I V(G)I2), Theorem 4 applies to infinite classes of graphs G with small edge density

(IE(G)I

_ constant.

I V(a)l)

and large diameter ( > constant" I V(G)I).

Denote by Lo the set of graphs of order at least 5 satisfying the conditions of Theorem 4.

Theorem 5 [4]. A graph G ~ L o is pancyclic if and only if G ~ .'Cir.

Theorem 6 [1]. A graph G e L o is Hamilton-connected if and only if it is 3-connected

and G q~ .,/!2.

We prove here that a graph G ~ Lo\¢~'1 has the following properties;

(a) For each pair of vertices x, y with d(x, y) > 3 and for each integer n, d(x, y) < n < IV(G)I - 1, there is an x - y path of length n.

(b) For each edge e which does not lie on a triangle and for each n, 4 < n < IV(G)I, there is a cycle of length n containing e.

(c) Each vertex of G lies on a cycle of every length from 4 to I V(G)I.

The last property implies the following: A graph G ~ L o \ J / 1 is vertex pancyclic if and only if each vertex of G lies on a triangle.

Theorems 2, 3 and 5 follow from our results.

2. Notations and Preliminary Results

Let P be a path of G. We denote b y / 7 the path P with a given orientation and by /5 the path P with the reverse orientation. If u, v ~ V(P), then u/Tv denotes the consecutive vertices of P from u to v in the direction specified by/7. The same vertices, in reverse order, are given by vPu. We use w + to denote the successor of w

(4)

Some Panconnected and Pancyclie Properties of Graphs 211 o n / ~ and w- to denote its predecessor. Also we denote by N(P) the set of vertices v outside P with N(v)A V ( P ) ~ ~. If W ~_ V(P) then W + = {w+/w e W} and W - = { w - / w ~ W}. We will say that a path P contains a triangle ata2aaal if ai, a2, a3 e V(P), ala3 e E(G) and a~ = a2 = a~'. A p a t h / 7 containing a triangle A is denoted by pa. T h e set of all triangles contained in/Ta we denote by T(/7~).

Analogous terminology is used with respect to cycles as well. We assume that an x - y p a t h / 7 has an orientation from x to y. A path (cycle) on n vertices will be denoted by P, (respectively, C,).

Let A and B be two disjoint subsets of vertices of a graph G. We denote by e(A,/3) the n u m b e r of edges in G with one end in A and the other in B.

Proposition 1 [9]. G e L o if and only if for any path uwv with uv ¢ E(G)IN(u) A N(v)[ > ]N(w)\(N(u) U N(v))l holds.

Corollary 1. I f G E L o then G is 2-connected and IN(u)A N(v)] _> 2 for each pair of vertices u, v with d(u, v) = 2.

Proof. Let d(u, v) = 2 and w e N(u) A N(v). T h e n u, v e N(w)\(N(u) U N(v)). There- fore, by P r o p o s i t i o n 1, IN(u) A N(v)l > 2. This implies 2-connectedness of G. [ ]

Proposition

2. Let G e L o and x, y be two distinct vertices of G with d(x, y) = l >_ 3. Then there exists an x - y path

P~+2.

Proof. Let Pl+l = UoUl... uz be an x - y path of length l where Uo = x and u~ = y. Since Pl+~ is an x - y path of m i n i m u m length then

N(ui) A V(Pt+l) = {ui_ l, ui+l } for each i = 1 . . . . , I - 1. (1) Suppose that no vertex v outside Pz+l is adjacent to two consecutive vertices of Pl+l- Then, since ui e N ( u o ) A N ( u : ) and u2 ~ N ( u i ) A N ( u 3 ) we obtain from (1), using P r o p o s i t i o n 1, the following:

d(u2) - 1 _> IN(uo)NN(u2)] >_ ]U(ui)\(U(uo)UU(u2))] = d(ul), (2) d(ul) - 1 >_ IN(ux)AN(u3)I > IN(uz)\(N(ul)U N(u3))l = d(u2). (3) But (2) contradicts (3). Hence there are vertices ui and v such that 0 _ i _< 1 - 1, v ¢ P~+I and vul, vui+l e E(G). This implies that there is an x - y path P~2 =

Uo ... u~vul+i.., uz of length I + 1 with d = uivu~+iuv []

Proposition

3. Let G e L o \ ~ l . Then each edge of G lies on a triangle or on a cycle C~.

Proof. Let G e L o and assume that there is an edge e of G which lies neither on a triangle n o r on a cycle C~. Let e = vlwl, d(wl) >_ d(vl) and N(wl) = { v l , . . . , v , ) . Clearly, v I vj ¢ E(G) for each j = 2, . . . , n because e does not lie on a triangle. Since d(vl, v2) = 2 there is a vertex w 2 e N ( v l ) A N(v2), w 2 ~ w i. Clearly, v2v j ~ E(G) for each j, 3 _< j _< n. (Otherwise e lies on a cycle C~.) Then, using P r o p o s i t i o n 1, we have

(5)

212 A.S. Asratian and G,V. Sarkisian Therefore n = d(vl) = IN(v1) O N(v2) I. Let N(vl) O N(V2) = {w 1 . . . wn}. Clearly, w~% ¢ E(G) for each pair i, j where 2 < i < j < n. (Otherwise there is a cycle C~ containing e.) Since d(wl,w~)= 2 then, using Proposition 1, we have IN(wx)ON(wi) I > IN(Vl)\(N(wi)UN(wl))l = [N(vl) I = n for each i = 2 . . . . , n. Therefore wiv.i e E(G) for each pair i, j, where 1 _< i, j _< n.

Let us show that V(G) = {vl .... , v., wl . . . w.}. Suppose that there is a vertex u e V(G)\{vl . . . v.,wl . . . %} such that uw~ e E(G) for some i, 2 < i < n. Then uv s (E E(G) for each j -- 2 . . . n. (Otherwise there is a cycle C~ containing e.) Since d(u, vl) = 2 then, using Proposition 1, we have

IN(u) fl N(vl)[ _> IN(wi)\(N(u) U N(vl))l >- n + 1

because u, v 1 .. . . . v. e N(w~)\(N(u)U N(vl)). But then n = d(vl) >_ [N(vOnN(u)l, a contradiction. Hence N(w~) = {vl .... , v.} for each i = i . . . n.

Using the same arguments it is possible to prove that N(vj) = {w~ . . . w.} for

each j = 1 , . . . , n. Consequently G = K.... []

Corollary 2.

L e t G e Lo\.//gl. Then each vertex of G lies on a triangle or on a

cycle C~.

Corollary

3. Each graph G e Lo \.//g 1 contains a triangle.

3. Main Resnlts

Proposition

4. Let G e L o and x, y be two distinct vertices of G. I f there is an x - y

path P, such that n = I V ( G ) [ - 1 and

IN(v)N{x,y}l <_ i for

the unique vertex v outside P, then there exists a Hamilton x - y path of G.

Proof. W.l.o.g. we assume v y ¢ E(G). Let W~ denote the set N(v)N V(P,) and p = I W~l. If vw + e E(G) for some w e W~ then there is a Hamilton x - y path. Suppose that vw+¢ E(G) for each w e Wo. Since n =

I V(G)l-

1 we have that S(v) n N(w +) ~_ W~ for each w e W~. Suppose that w+g + ¢ E(G) for each pair w, g e W~. Then IN(v) N N(w+)[ _< e(Wv, Wv +) weWu and ~, IN(w)\(N(v)UN(w+))l > e(W~, W~) + p w e W v

since v E: N(w)\(N(v)U N(w+)) for each w e W~. By Proposition 1 we have IN(v)NN(w+)I > ~ IN(w)\(N(v)NN(w+))I.

weW~ w~W~

But the last inequality contradicts the previous two. So w+g + e E(G) for some pair of vertices w, 9 where w occur on P. before 9. Then there exists a Hamilton x - y

(6)

Some Panconnected and Pancyclic Properties of Graphs 213 T h e next technical l e m m a plays a key role in the proofs of the s u b s e q u e n t theorems.

L e m m a . L e t G ~ L o and x, y be two distinct vertices o f G. I f there exists an x - y path P f such that 4 <_ n <_

I V(G)I

- 2 and IN(v) fq {x,

y}l

-< 1 f o r each v ~ V ( G ) \ V ( P f ) then there exists an x - y path P~2-t such that 1 < t < 2 and V ( P f ) c V(Pf2r).

T h e p r o o f o f the l e m m a will be given later.

T h e o r e m 7. L e t G ~ L o and e = x y be an edge o f G which does not lie on a triangle. Then e lies on a cycle C, f o r each n, 4 <_ n <_

I V(G)l.

Proof. It is sufficient to p r o v e that there exists an x - y p a t h P, for e a c h n, 4 _< n < I V(G)[. Since e does n o t lie o n a triangle then, b y P r o p o s i t i o n 3, e lies o n a cycle C~. H e n c e there is a n x - y p a t h Pc and an x - y p a t h P~. S u p p o s e t h a t there exist x - y p a t h s P4 . . . . , P,-1 a n d a n x - y p a t h P~ for s o m e n, 5 < n <

I V(G)l

- 1. If n = IV(G)1 - 1 then, b y P r o p o s i t i o n 4, there exists a H a m i l t o n x - y p a t h . If n _< I V(G)t - 2 then, b y the l e m m a , there exists an x - y p a t h P ~ t where 1 _< t _< 2. I f t = 2 a n d ,41 = w - w w + w - then we can o b t a i n a n x - y p a t h P,+I b y deleting the vertex w f r o m P~.~ 2.

Repetition o f o u r a r g u m e n t shows that there is an x - y p a t h P, for e a c h n,

4 < n < [V(G)]. [ ]

U s i n g P r o p o s i t i o n 2 instead o f P r o p o s i t i o n 3 and the s a m e a r g u m e n t s as in the p r o o f of T h e o r e m 7, we c a n p r o v e the following.

T h e o r e m 8. L e t G ~ L o and x, y be two distinct vertices o f G with d(x, y) > 3. Then f o r each n, d(x, y) + 1 < n < I V(G)I, there exists an x - y path P~.

Let a cycle (~ c o n t a i n a triangle d = ala2asa~. W e call the vertex a2 a centre of ,4 a n d d e n o t e it b y s(,4).

Proposition

5. L e t G ~ L o \ J l [ 1. Then each vertex g o f G lies on a cycle C~ or on a cycle C~ such that g ~ s(`4).

Proof. If g does n o t lie o n a triangle then, by C o r o l l a r y 2, g lies on a cycle C5 a a n d g ~ s(A). N o w s u p p o s e t h a t g lies o n a triangle x g y x .

Case I. d(g) = 2. Since ] V(G)] > 5 and, by C o r o l l a r y 1, G is 2-connected then m u s t exist a vertex v with d ( g , v ) = 2. Clearly, I N ( g ) N N ( v ) ] > 2. Since d ( g ) = 2 then N(g) f3 N(v) = {x, y}. So we h a v e a cycle C~ = x v y g x with d = x v y x a n d s(d) = v. Case 2. d(g) > 3 a n d g has a c o m m o n neighbour with x or y outside d.

W.l.o.g. s u p p o s e xv, vg ~ E(G) for some v ~ {x,y,g}. T h e n there is a cycle C~ = x v g y x with d = x v g x a n d s(A) = v.

Case 3. d(g) > 3 a n d g has no c o m m o n n e i g h b o u r with x and y outside d.

Consider a vertex v ~ N ( g ) \ {x, y}. Since d(v, x) = 2 we have that IN(x) N N(v)] > 2 a n d there exists a vertex z ~ N(v) fq N(x), z ¢ g. N o w we have a cycle C~ = x y g v z x

(7)

214 A.S. Asratian and G.V. Sarkisian

Theorem

9. Let G ~ Lo\.//lt. Then for each vertex g of G and for each n, 4 ~ n <_

[ V(G)I, there is a cycle of length n containing 9.

Proof. By Proposition 5, 9 lies on a cycle C, a where s(A) # g and 4 _< r _< 5. Hence there is a cycle C4 containing 9.

Suppose there exist cycles C4 . . . C., 4 _< n _ J V(G)[ - 1, containing 9 such that the last cycle C. contains a triangle d = ata2aaa 1 with a 2 = s('4):~ g i.e. C. = C. a. If there exists a vertex v ~ N(Ca,) which is adjacent to two consecutive vertices w and w + of C. n then there is a cycle C~,+ta' = w v w + ~ w with ,41 = wvw+w

where s(zil) = v # g. Now let no vertex v ~ N(C~.) be adjacent to two consecutive vertices of C. ~. Consider a vertex v ~ N(Cn.) and the triangle ,4 = a l a2 a3 a l. Clearly,

9 v~ a2 .

Let us define vertices x and y, a triangle Ao and an x - y path P.~o in the following way.

I f g ~ Wo and g # a a then x = g, y = g - , d o = ,4 and P,~° = g~ag-.

I f g ~ W~ and g = a3, then x = 9, Y = g+, do = a3a2ata3 and Pro = 9C~.9 +.

I f g ¢ W~ and O # at then x = g+, y = 9, Ao = d and pro = g+~/,g.

I f o ¢ W~ and O = at then x = g - , y = O, do = a3a2atas and P.~o = 9-C~.g.

Clearly, IN(v) fl {x,y}[ < 1 for each v ~ N(pfo). I f n = [ V(G)[ - 1 then, by Prop- osition 4, there exists an Hamilton x - y path. Clearly, since x and y are adjacent, there also exists a Hamilton cycle of O. If n _< [ V(G)[ - 2 then, by the lemma, there is an x - y path Pfct such that 1 < t < 2 and V(Pf °) = V(Pf~.,). Since g ~ {x, y} then 9 # s(,41). The path P.a~t define a cycle C,4)t. I f t = 2 we can obtain a cycle C,+I containing g by deleting the centre s('41) from C.n~ 2.

Repetition of our argument shows that there is a cycle C. for each n, 4 _< n <

I V(G)[. [ ]

Clearly, Theorem 5 follows from Theorem 9 and Corollary 3. Using T h e o r e m 9 we can formulate a criterion for a graph G e Lo to be vertex pancyclic.

Theorem

10. A graph G ~ L o is vertex pancyclic if and only if every vertex of G lies

on a triangle.

Corollary 5. Let G be a connected graph of order at least 3 where d(u) + d(v) > IN(u)U N(v)U N(w)I + 1 for any path uwv with uv q~ E(G). Then G is vertex pan- cyclic if and only if every vertex of G lies on a triangle.

Let us point out that for each p _> 7 the graph K ] v (Kt + Kp-4) satisfies the condition of Corollary 5 but contains a vertex that does not lie on a triangle. The next result immediately follows from Theorem 9.

Corollary

6. Let G ~ Lo\u/t'l be a graph of order p >_ 5 where d(u) + d(v) >_ p for

each pair of vertices u, v with d(u, v) = 2. Then each vertex of G lies on a cycle of every length from 4 to p.

Finally we give the proof of the lemma.

(8)

Some Panconnected and Pancyclic Properties of Graphs 215 we d e n o t e by W~ the set N(v)A V(P~). Let U1 = {veN(P~a)/IW~l =

1}

a n d U2 =

N(P~)\U1.

S u p p o s e t h a t there does n o t exist a n y x - y p a t h P ~ such t h a t 1 _< t _< 2 a n d

V(P~) ~ V(P~.,). T h e n the following properties hold. P r o p e r t y 1. vw + ¢ E(G) for each v e N(P~) and each w e W~.

Property 2. I f v e U1, W~ = {w} and w ~ {x, y} then the set T(ff~) contains the unique triangle w- ww + w -.

Proof. Let aia2a3a 1 be a triangle f r o m the set T(ff~a). S u p p o s e a2 # w. Since

d(v, w-) = 2 = d(v, w +) then, b y C o r o l l a r y 1, there exist vertices vi a n d v2 such th~it

vl e (N(v) A N ( w - ) ) \ V(P~) a n d v 2 e (N(v) A N(w+))\ V(P~a). This gives a n x - y p a t h

~xff2aw-vivwff~ay ifa2ew+ff~ay

with zt t = aia2a3at such t h a t V(P. a) ¢ V(P~a.~2), a contradiction. [ ]

Property 3. []2 ~ ~ .

Proof. S u p p o s e t h a t U 2 = Z . L e t v e U l, W~ = (wl} and A 1 = aia2a3a i be a trian- gle f r o m the set T(P~a). T h r e e cases are possible: wi = x, w i y and w i ~ {x, y}.

Let wl = x. Since d(v,w~)= 2 a n d IW~l = 1 then, b y C o r o l l a r y 1, there is a vertex v 1 e (N(v) A N(w~))\ V(P~a). Hence, b y P r o p e r t y 2, a 1 = w l, a 2 = w~ a n d

aia2a3a 1 is the unique triangle in the set T(P~). Since v 1 e U1, d(vl,a3) = 2 a n d

I Wo~l = 1 then, b y C o r o l l a r y 1, there is a vertex v2 e(N(vt)AN(a3))\V(P~).

Clearly, v 2 e U i, Wv: = {a3} a n d a2a~ q~ E(G). This contradicts P r o p e r t y 2. U s i n g the s a m e a r g u m e n t s we o b t a i n contradictions in the cases w = y a n d

w ~ (x, y}. Hence, U 2 ~ ~ . [ ]

P r o p e r t y 4. Let o e U 2, vy q~ E(G) and Q c_ W~ = (wl,..., wp}. Then

IN(v)AN(w~)I > ~ IN(w,)\(N(v)U N(w~~)I (4)

w~Q w~Q

Furthermore, if a 1 a2a 3 a 1 is a triangle from T(fff) with {a l, a 2 }A Q = ~ then

N(v) A N(w~') ~ W~ for each w i e Q (5)

and

w~ w~ q~ E(G) for each pair of vertices w i, wj e Q (6)

Proof. Since d(v, w~) = 2 for each wi e Q then (4) follows f r o m P r o p o s i t i o n 1. I f (5) does not hold then there exist a vertex v i e (N(v) A N(wi ))\ Wv + for s o m e wi e Q a n d an x - y p a t h Pf-~2 = xfffwivviw~fffY with A1 = ala2asal, a contradiction. So, (5) m u s t hold.

If w~ w f ~ E(G) for s o m e p a i r w~, wj e Q, i < j, then there is a n x - y p a t h

(9)

216 A.S. Asratian and G.V. Sarkisian

~1 = ~ ala2aaal

~a3a2ala3

a contradiction. So, (6) holds~

if a l ¢ w~/~ffw~ otherwise

[] Property 5. Let v e U z and vy q~ E(G). I f ala2a3al is a triangle from the set T(ffn a) then WvA {al,a2} ~ ~.

Proof. Let wl . . . w, denote the vertices of Wo occurring on/~,~ in the order of their indices. Suppose {ax, a2} r) ~ = ~. Set Q --- W v. Then, by Property 4, we have (4), (5) and (6). Furthermore, we have

~. IN(v)AN(w~')I < e(Q,Q +) (7)

V~t 6 Q

and

IN(w,)\(N(v)t.J N(w~'))l > ~(Q,Q+) + p (8) because v ¢ Q+ and v e N(wi)\(N(v)UN(w~)) for each i = 1, ..., p. But (7) and (8)

contradict (4). So, {al,a2} CI W v # ~. []

Property 6.

I Wvl

= 2 for each vertex v e U2. Moreover w-w+ e E(G) for each

w

(x, y}.

Proof. Let v e U2. Then I W~ A {x, Y}I < 1 by the assumption (in the lemma). W.l.o.g. we may assume that vy d~ E(G). Let wt .... , wp be the vertices of W~ occurring on/~ff in the order of their indices. By Property 5 we have that W~ fl {al, a2 } # Z for each triangle aia2a3at from the set T(ff~). Let k be the minimum i, 1 < i < p, for which w ; w : e E(G) or w,w ++ e E(G).

Case 1. k < p.

Let ala2aza , be a triangle with wke(al,a2}. Set Q = W~\{wk}. Since Q f] (al, a2} = ~ then by Property 4 all of (4), (5) and (6) hold. Since the vertices w k and w~ + can be adjacent for each wi e Q, we have

IS(v)AN(w~')l < s(Q,Q +) + p - I. (9)

Wl6Q

If w~+l w~+ 1 ¢ E(G) then

IN(wi)\(g(v)U N(w~'))l > e(Q,Q +) + p (10)

WiG~

because w~+ 1 ¢ Q+, w~+ 1 e N(Wk+i)\(N(v ) U N(w~+l) ) and v d~ Q+, v e N(wi) \ (N(v)UN(w~)) for each w~ e Q.

But (9) and (10) contradict (4). Consequently w~+lw~+ 1 e "E(G). Moreover note that w~" # w~'+ 1, since otherwise we find an x - y path of length n + 1 containing a triangle. N o w set Q = W~\ {wk+ x }. Then, by Property 4, all of (4), (5) and (6) also hold for the new Q. Clearly, w~'w~+i ¢ E(G) for each j # k, 1 < j < p. (Otherwise there is an x - y path P~_~ with/t 1 = ala2a3al, a contradiction.) Hence, we have

IN(v)AN(w~)I <_ e(Q,Q +) + 1. (11)

(10)

Some Panconnected and Pancyclic Properties of Graphs 217 Furthermore,

IN(w,)\(N(v)UN(w?))I > e(Q,Q ÷) + p - 1 (12) wi~Q

because v ~ Q+ and v ~ N(w,)\(N(v)U N(w~)) for each w i e (2. It follows from (11), (12) and (4) that p = 2. So, k -- 1, p = 2 and w~w~" s E(G).

If w 1 ~ x then we can obtain w[w~ ~ E(G) by considering a y - x path P~ and using the above arguments.

Case 2. k = p.

We assume that x = w 1. (Otherwise we can consider a y - x path iP~ and use the same arguments as in Case 1). Set Q = {w 1 .. . . . wv_l}. Let alazasal be a triangle with wve {al,a2}. By Property 4 all of (4), (5) and (6) hold because Q Cl {al, a2} = N. Furthermore, since the vertices wp and w~ can be adjacent for each w~ s Q, (9) also holds. On the other hand the inequality (12) also holds, because v ~ Q+ and v ~ N(w~)\(N(v) U g(w~)) for each wi ~ Q. It follows from (9), (12) and (4) that

Y'. Ig(v)NN(wi~)l = e(Q,Q+) + p - 1 (13)

wi6Q

This implies that w~+wpsE(G) for i = 1 .. . . . p - 1 . Now we have that

p >_ IS(v)Ag(w~')l > Ig(wp)\(N(v)UN(w~))l and {v,w~" .. . . . w~_~} ~ N(w,,)\ (N(v) U N(w~)). Therefore, for each i = 1 .. . . , p - 1 we have

N(wp)\(N(v) IJ N(w+)) = {v, w-~, . . . . wp_ 1 *

}

(14) Since w~ e N(wp)\N(v), (14) implies that

w~'w~ ~ E(G) for each i = 1 .. . . . p - 1 (15) Suppose that w;w~ ~ E(G). Then, by the definition of p, wpw~÷s E(G). Since

w~ + ~ N(wp)\N(v), (14) implies that w~w; ÷ ~ E(G). By (15), we also have that

w~w; ~ E(G). Then G has an x - y path P~4t = xvwpP~w~w~ff~y with A~ =

w~w;w;÷w~; a contradiction. So, w ; w ; ~ E(G) and, therefore, w~_l ~ w;. More- over, (14) implies that w'~w~ ~ E(G) since w~ ~ N(wv)\N(v). + ,.- + -.

I f p > 3 t h e n G h a s a n x - y p a t h P ~ ! ~ - - xvwp_lP;, w 1 u,~ P~ wi,_twpP;, y a - ~ a with

d~ = w~_~ wpw~ w~_~; a contradiction. Therefore, p = 2 and w~ w~ ~ E(G). []

Now using the properties above we will obtain a contradiction. Let v ~ U2, W~ = {wl, w2} and assume that w 1 occur o n / ~ before w2. W.l.o.g. we may assume that vy ~ E(G). Then, by Property 6, w~w~ ~ E(G). Clearly, w~w 2 ~ E(G) because IN(v) fl N(w~)l > 2, N(v) Cl N(w~) c_ W~ and I W~I = 2. Since G is 2-connected and n < IV(G)] - 2 we have that N(Pf)\{v} ~ ~. Let us show

uw[ ~ E(G) for each u ~ N(Pf)\{v} (16)

Suppose uw; e E(G) for some u ~ N(Pf)\{v}. Then, by Property 2, u ~ Uz. F o r the vertex g = w; we have g - ~ wt and g ~ W,\{x,y}. Hence, by Property 6,

g-g+ ~ E(G). But then for the triangle g - o g ÷ g - we have {g-,g} fl ~ = 2~. This contradicts Property 5. So, (16) is proved.

(11)

218 A.S. Asratian and G.V. Sarkisian Now let us show that

uw~ ¢ E(G) for each u ~ N(P~)\ (v} (17)

Suppose that uw2 ~ E(G) for some u ~ N(P~)\{v~. Clearly, uv ~ E(G). (Otherwise there exists an x - y path Pff-~2 = xP~wlvuwzw~ff~w2w~'~Y with zt 1 = vuw2v, a

contradiction.)

We have w2, v, u e N(w2)\(N(w2) O N(v)) and d(w~, v) = 2. Therefore, by Pro- position 1, IN(w2) fq N(v)l >_ 3. Since I W~l -- 2 there is a vertex vl e (N(v) 13 N(w~))\ W~

which contradicts (16). So (17) is proved.

Consider a vertex u ~ N(P~)\(v}. By (17) and Property 2 we have u ~ U2. Using Property 5 with the triangle w~w2w~w z, the vertex u and the p a t h / ~ we obtain that uy~E(G). Then uxq~E(G) because IN(u)13{x,y}l< 1. Let W~= {gl,gz}

where g2 = Y. Then, by Property 6, #-ig'~ ~ E(G).

Using Property 5 with the triangle + w2 w2w2 w2, the vertex u and the path PA, - + '-~ we obtain uw~ ~ E(G). Suppose that w~" # y. Then gl = w~" and g'~g~ ~ E(G). This

W - +

implies 2 01 ~ E(G). (Otherwise w; , g~, u ~ N(gl)\(N(u ) U N(g'~)) and, by Propo- sition 1, IN(u) f) N(g~)l > 3. Since I W~l --- 2 ~ e r e exists a vertex u 1 ~ iN(u) 13 N(#~'))\ W~ and an x - y path Pf-~2 = xPfgi uulg'~Pfy w i t h a l = w;w2w~w~, a contradic- tion.) But now we obtain an x - y path Pf21 = xP~w~vw2w'~Pfw;glg~PfY with

Zl l --- w; gl g~ w; , a contradiction.

So, w~" = #z = Y. Then using Property 5 with the triangle g~'g~g~'g~', the vertex v and the x - y path/~f, we obtain w 1 ~ {#~',gl}. If gl = wl then N(v)nN(w~) = {w~,w2} , {u,v,w?} ~_ N(w~)\N(w~) and v, w~ ~ N(wt)\(N(v)NN(w~)) imply

uv ~ E(G). But then there is a path Pf~1 = xfffw~uvw2w~fffw~y with A2 = w~ uvw~,

a contradiction. So, w~ = g~.

N o w we have: w~ = gl, w~ ---- y = #2, gZg~ ~ E(G), w~w~ ~ E(G), W'~W 2 ~ E(G), d(w l, u) = 2 and w~ ~ N(wl) 13 N(u). Clearly, N(u) 13 N(Wl) ~- W~. (Otherwise there exists a vertex ul ~ (N(u) 13 N(wl))\ W,, and an x - y p a t h / ~ 2 : = xP~wl u~ uw'~P~y

with d~ = w~ w z w~" w~, a contradiction.) Hence, using Proposition 1, we obtain 2 > IN(wx) O N(u)l > IN(w~)\(N(Wl)U N(u))l (18) On the other hand, we have wlu, uwz ~ E(G) and u, wl, wz ~ N(w~). Then wl wz E(G). (Otherwise u, wl, w2 ~ N(w~)\(N(u) t3 N(wl) ) which contradicts (18).) Now we obtain an x - y path P~;1 = xff~wl vwz w~ ff~w;y with .41 = wl vwz w,, a contra- diction. The proof of the lemma is complete.

Acknowledgement. We thank R. H~iggkvist for discussions and many helpful comments. We also thank the referees for their remarks and suggestions which led to this improved version.

References

1. Asratian, A,S.: A criterion for some hamiltonian graphs to be Hamilton-connected. Australasian J. Comb. 10, 193-198 (1994)

2. Asratian, A.S., Broersma, H.J., van den Heuvel, J., Veldman, H.J.: On graphs satisfying a local Ore-type condition. J. Graph Theory 21, 1-10 (1996)

(12)

Some Panconnected and Pancyclic Properties of Graphs 219

3. Asratian, A.S., Khachatrian, N.K.: Investigation of graph's hamiltonicity using neigh­ borhoods of vertices (Russian). Doclady Acad. Nauk Armenian SSR 81, 103-106 (1985) 4. Asratian, A.S., Sarkisian, G.V.: On cyclic properties of some hamiltonian graphs (Rus­

sian). Diskretnaja Matematika 3, 91-104 (1991)

5. Bondy, J.A.: Pancyclic graphs 1. J. Comb. Theory Ser. B 11, 80-84 (1971)

6. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, MacMillan, London and Elsevier, New York

7. Xiaota, Cai: On the panconnectivity of Ore graph. Scientia Sinica, 27, 684-694 (1984) 8. Hiiggkvist, R., Faudree, R.J., Schelp, R.H.: Pancyclic graphs-connected Ramsey number.

Ars Combinatoria 11, 37-49 (1981)

9. Hasratian, A.S., Khachatrian, N.K.: Some localization theorems on hamiltonian cir­ cuits, J. Comb. Theory Ser. B 49, 287-294 (1990)

10. Ore, 0.: Note on hamiltonian circuits. Amer. Math. Monthly 67 55 (1960)

References

Related documents

The aim of this study is to investigate whether differences in personality can be assessed in a group of eight captive bottlenose dolphins (Tursiops truncatus) based on

Det dikten ”A Sponge Full of Vinegar” understryker är idéer som främst kan härledas till katolsk teologi såsom den kommer till uttryck hos främst Bonaventura –

Arbetet med Kanban gjorde istället att utvecklarna kunde fokusera på färre uppgifter samtidigt, vilket ökade kvaliteten (Nikitina et al. 29) menar att projekt ”byråkratiseras”

Sensitivitet, specificitet och noggrannhet presenterad i procent i jämförelse och i kombination mellan biomarkörer och modaliteter för att diagnostisera Alzheimers sjukdom

Supreme Court, in a widely publicized case, is deciding on the states’ duty to recognize lawfully licensed same-sex marriages ( Obergefell v. Hodges , judgment expected at the end

For each ontology in the network, we want to repair the is-a structure in such a way that (i) the missing is-a relations can be derived from their repaired host ontologies and for

This report explores the results of respondents from the Nordic countries (Denmark, Finland, Norway and Sweden) in order to understand if their perspectives regarding

Keywords: Quadrature domain, Two- phase problems, Uniqueness, Free boundary prob- lem, Level set method, Shape optimization.. Postal address: Department of Mathematics