**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

**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.*

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 *

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

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

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 *

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.

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

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 **

**w**

**(x, y}. **

**(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)

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.

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)

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)**