Some panconnected and pancyclic properties of
graphs with a local ore‐type condition
Armen S. Asratian and G. V. Sarkisian
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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.
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Some Panconnected and Pancyclic Properties of Graphs
with a Local Ore-type Condition
A.S. Asratian
l.
2and G.V. Sarkisian
21
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
uwvwith
uvff
E(G), .where
N(x)is the neighborhood of a vertex
x.We prove that a graph
Gwith this condition, which is not complete bipartite, has the
following properties:
a) For each pair of vertices
x, ywith 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
ewhich does not lie on a triangle and for each
n,4 :;;;
n:;;; I V(G)I, there
is a cycle of length
ncontaining
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
Glies on a triangle.
1. IntroductionWe use Bondy and Murty [6] for terminology and notation not defined here and
consider finite simple graphs only. For each vertex
uof a graph G we denote by
N(u)the set of all vertices of
Gadjacent to
u.The distance between vertices
uand
visdenoted by
d(u,v). A path with
xand
yas end vertices is called an
x-
ypath.
An
x-
ypath is called a Hamilton path if it contains all the vertices of G. A graph
Gis Hamilton-connected if every two vertices of
Gare 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
yand for each I, d(x, y) ::::; / ::::;
p- 1, there is an x -
ypath
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. Gis called vertex pancyclic (edge pancyclic) if each vertex
(edge) of G lies on a cycle of every length from 3
topinclusive.
Let M
1= {K
n,
n,n;;::: 2} and J1
2= {G: K
n,
ns;;; G s;;; K
n VK�,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 pathP~+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 acycle 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 - ypath 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 vsince 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 lieson 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 foreach 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 eachw
(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 thatw~'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.
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