A Cnjectue n Dminating Cycles
Bent N. Clak
Deatment f Cmute Science
Univesity f Watel
Watel, Ontai, CANADA
Chales J. Clbun
Deatment f Cmute Science
Univesity f Watel
Watel, Ontai, CANADA
Paul Edös
Mathematical Institute
Hungaian Academy f Sciences
Budaest, HUNGARY
ABSTRACT
A dminating cycle in a gah is a cycle in which evey vetex f
the gah is adjacent t at least ne vetex n the cycle . We cnjectue
that f each c thee is a cnstant k c such that evey c-cnnected gah
with minimum degee 6 >
+1
+k, has a dminating cycle . We shw
that this cnjectue, if tue, if best ssible . We futhe ve the cnjectue f gahs f cnnectivities 1 thugh 5 .
1 . Intductin
F ntatin, usually we fllw Bndy and Muty [l[ . The numbe f vetices, the
cnnectivity and the minimum degee ae dented by n, c and 6, esectively . A dminating cycle is a cycle L in gah G f which evey vetex f C is adjacent t at least
ne vetex f L . A me secific tye f cycle is a D-cycle , which is a cycle L in gah
G f which evey edge f G is incident t at least ne vetex f L .
Dminating cycles have been studied fm an algithmic viewint [3, 4 and 71
with alicatins in netwk design in mind . We ae inteested hee instead in studying
an extemal blem, namely the minimum degee which ensues that a c-cnnected
gah cntains a dminating cycle . Ou imay mtivatin is nt algithmic, but
athe t extend evius eseach n D-cycles and Hamiltn cycles . A D-cycle can he
cnsideed as a genealizatin f a Hamiltn cycle and a dminating cycle a genealizatin f a D-cycle . Theefe the smallest minimum degee that guaantees a dminating
Cngessus Numeantium, 47 (1985), .189-197
cycle shuld be smalle than that f a D-cycle, which in tun shuld be smalle than the
sufficiency cnditin with esect t 6 f Hamiltn cycles .
Diac's classical esult gives the sufficiency cnditin with esect t 6 f Hamiltnicity [5] .
Theem Al . Let G be a gah with n>3 and 6>
2. Then G is Hamiltnian .
F D-cycles, a theem f Nash-Williams (see [2]) establishes an ue bund, which an
examle f Veldman [8] shws is best ssible :
n3
1 . Then G has a DTheem A2 . Let G be a c-cnnected gah (c>2) with 6 >
cycle .
Bth these esults give sufficiency cnditins with esect t 6 deending nly n n . As
lng as the cnnectivity is high enugh, it is ielevant .
Befe we ve esults abut dminating cycles, we need a lemma, which elies n
the fllwing tw theems . Bndy [2] gives Theem B, which elates cnnectivity,
minimum degee and what can lie ff a lngest cycle . A gah is n-ath-cnnected if
any tw vetices ae cnnected by a ath f length at least n .
Theem B . Let G be a c-cnnected gah such that the degee-sum f any c+1
indeendent vetices is at least n+c(c-1), whee n>3, and let L be a lngest cycle in G .
Then G-L cntains n (c-1)-ath-cnnected subgah .
Theem C, fm Edös and Gallai [6], elates numbe f edges and the length f the
lngest cycle .
Theem C . Let G be a gah n n vetices with at least
2
d(n-1)+1 edges, whee
d>l . Then G cntains a cycle f length at least d+l .
Lemma 1 fllws diectly fm these theems .
+c-1 and let L be a
-lngest cycle in G . Then all subgahs H in G-L have less than (c-2)(v(H)-1)+1
edges .
Pf. Let H be a subgah f G-L . Fm Theem C, H is nt (c-1)-ath cnnected, which imlies n cycles f length 2c-3 me since such a cycle is (c-1)-athLemma 1 . Let G be a c-cnnected gah, c>3, with 6 >
cnnected . Using Theem D, H must have less than 2(2c-4)(v(H)-1)+1 edges. 0
2 . Dminating cycles in gahs with small cnnectivity .
Ou gal is t establish a sufficiency cnditin f the existence f dminating
cycles . In de t establish a geneal atten, we begin by ving sufficiency cnditins
with esect t 6 f dminating cycles in gahs with small cnnectivity . Late we
extalate this atten t fmulate a cnjectue abut the sufficiency cnditin .
190
Lemma 2 . Let G be a cnnected gah with n>3 and b >-! . Then G cntains a dminating cycle .
Pf. Fm Theem A, G has a Hamiltnian cycle . A Hamiltn cycle dminates . 0
Lemma 3 . Let G be a 2-cnnected gah with n>3 and b>3 . Then G cntains a
dminating cycle .
Pf. Fm Diac [5), G has a cycle L f length at least
3
. Since b
> 3
,
evey ve-
tex must have a neighbu n the cycle . Until this int, the situatin f dminating cycles is essentially the same as it is f Dcycles ; f c>3, hweve, we see a adical diffeence .
Theem 1 . Let G be a 3-cnnected gah, with sufficiently lage n and 6
> 4 +2 .
Then G has a dminating cycle .
Pf. Let L be a lngest cycle in G . Fm Diac [5], L has length at least 26 . If L
des nt dminate then thee exists sme vEV(G) such that V(H)fV(L)=O, whee
V(H)=N(v) and v(H)>b . By lemma 1,
e(G-L) < (c-2)(v(G-L)-1)+1 <
-4 .
2
Thee must exist sme x,yEV(H) such that dG _L (x) < I and dc-L(y) < 1 (see Figue 1) .
Figue 1 .
191
Othewise d G _L (u) > 2 f all uEV(H) excet f ssibly sme x'EV(H) . By lemma 1,
n uEV(H) can have neighbu wEV(H), since then e(u+v+w) = 3 . Theefe each edge
must accunt f ne vetex degee and since v(H) > 6
E dc-L(x) <
e(G-L)
2(-!!-+2-1)
u E:V(H)
ufz'
Then
d L+z (x)
> 4 +1
and
dL+y (y)
> 4 +1 .
= 2 +2 > 2 -4 .
The neighbus f x and y n L must be
at least fu aat n L we culd fm a lnge cycle by including x,v,y and mitting
the vetices n the cycle between the neighbus f x and y . Theefe
v(L) > 4(4 +1) > n . L must dminate .
ED
Theem 2 . Let G be a 4-cnnected gah, with sufficiently lage n and 6
> 5 +3 .
Then G has a dminating cycle .
Pf. Let L be a lngest cycle in G that dminates the mst vetices . Again L has
length at least 26 . If L des nt dminate then thee exists sme v and H as in theem
1 . By lemma 1,
e(G-L) < 2(5 n-6-1)+1
= 5n-13 .
Thee must exist sme x,yEV(H) such that d G _ L (x) < 11 and d G _L (y) < 11 . Othewise
dc-L(u) > 12 f all uEV(H) excet ssibly f sme x'EV(H) . Since each edge can
accunt f tw vetex degees and v(H)>b
I E dG-L(x) < e(G-L) 2 .EV(H)
1 12( n5 +3-1) = 65 n+12
2
< -%-13 .
5
u#z'
The neighbus f x and y n L must be at least fu aat we culd fm a lnge
cycle . Cnside any tw neighbus f x and y n L that ae fu aat, 1 1 and l s , and
the vetices between them n L, 1 2,1 3 and 1 4 (see Figue 2) . All neighbus f 13, the
than 1 2 and 1 4 , must nt be n L we culd cnstuct a cycle f equal length that dminates ne me vetex by leaving 12,13,14 ff the cycle and including x,v,y . This wuld
cntadict the chice f L . Theefe if any neighbus f x and y n L ae fu aat,
v(L) > 4(6-11)
= 6 n-32 and v(G-(L+H+v))
> 6-2
= 5 +1 . But then G must have
me than n vetices . If the neighbus f x and y n L ae all at least five aat, then
v(L) > 5(6-11) = n-40 and again we have me than n vetices . Theefe L must
dminate . Theem 3 . Let C he a 5-cnnected gah, with sufficiently lage n and b
> 6 +6 .
Then C has a dminating cycle .
Pf . Let L he a lngest cycle in G that dminates the mst vetices . Again L has
length at least 26 . If L des nt dminate then thee exists sme v and H as in theem
I and 2 . By lemma 1,
e(C-L) < 3(
3 n-12-1)+1
= 2n-38 .
simila t theem 2, thee must exist sme x,yEV(H) such that dG _ L (x) < 23 and
192
Figue 2 .
d G _ L (y) < 23 . The neighbus f x and y n L must be at least fu aat we culd
fm a lnge cycle . Cnside any tw sets f vetices f L that have tw neighbus f
x and y fu aat, 11,12,13,14,15 and lk,lk+i,lk+2,lk+3,lk+4 (see Figue 3) .
Figue 3 .
1 93
Bth 1 3 and 1k+2 must have all neighbus ff the cycle, excet f thei immediate neighbus n the cycle . These neighbus must als be disjint we can fm a lnge cycle
as indicated in Figue 3 . Theefe if tw me sets f neighbus f x and y ae fu
aat, v(L) > 4(6-23)
= 6 n-68 and
v(G-(L+H+v)) > 2(b-2)
= 6n+8 .
as me than n vetices . If thee is at mst ne set f neighbus f
aat then v(L) > 5(6-23)-1
= 6 n-86 .
x
But then G
and y that is fu
With this new estimate f the size f v(L)
by
lemma 1,
e(G-L) <
3(6 +86-1)+1 = 2 +256 .
Simila t theem 2, thee must exist sme x,yEV(H) such that dG _L (x) < 6 and
d-L(y) < 6 . If thee is at mst ne set f neighbus f x and y that is fu aat then
v(L) > 5(b-6)-1
inates .
=
6
n-1 and we again have me than
n
vetices . Theefe
L
dm-
0
3 . The cnjectued sufficiency cnditin
Even thugh we d nt knw the exact esult f highe cnnectivity, the fllwing
examle laces a lwe bund n the sufficiency cnditin . Let c>1, A>c and G cnsist
f the fllwing subgahs :
X = K,
Yi = KA V z„ i = 1,2,
c+1,
with exta edges fm evey vetex in X t evey vetex in Y i -zi (see Figue 4) .
~
'
00
is
Z,
0
0,-"
1
Z2
Figue 4 .
G has cnnectivity c, b = A
= c+1
n -(1+ nc+1)
and n dminating cycle . F a dm-
inating cycle t exist each Yi must have at least ne vetex n the cycle s each z i will
be adjacent t the cycle, but t include each Yi we need c+1 vetices in X . G shws
that the sufficiency cnditin is geate than
n -2 .
c+1
1 94
We cnjectue the fllwing sufficiency cnditin f dminating cycles in tems f
6 and c :
Cnjectue 1 . Let G be a c-cnnected gah with n>3 and 6 >
c+1
+k,, whee kc is
a cnstant deending nly n c . Then G has a dminating cycle .
Cnjectue 1 is the best ssible by the evius examle s it may allw values f
6 that ae t lw t guaantee a dminating cycle . We ae much me cetain that the
sufficiency cnditin f 6 is nt a cnstant, like
, as it is f Hamiltn cycles and D6
cycles . It may be me easnable t ty t find a sufficiency cnditins, in tems f 6,
f each c that ae less than
and decease as c inceases .
6
One he in ving such a cnjectue is t shw that when thee is a dminating
cycle, sme lngest cycle dminates as we did in theems 1,2 and 3 . Hweve, the fllwing examle shws that lngest cycles ae nt necessaily dminating althugh dminating cycles exist . Given c>6 and m>6 we cnstuct such a gah G with 6 = n64
n n=6m+2 vetices . Let G cnsist f the fllwing subgahs :
H=vVH'
whee v is a vetex and H' is a K m , and
m
J=KmV UY'
i-l
whee Y=K 4 ) with exta edges fm evey vetex in H' t evey vetex in the K m (see
Figue 5) .
n 4
Then G has cnnectivity c and 6 = m+1 = 6 . A lngest cycle in G has all f the
vetices f each Y,, i = 1,2, • • • m, and als K m ; t include H wuld add 3 vetices,
but wuld als emve a Y fm the cycle theeby subtacting me than 3 vetices . N
lngest cycle is dminating, but a dminating cycle exists . F 6 >
c+1
+k,, sufficiently
lage n, and c>
_ 6 such examles exist s f highe cnnectivity we cannt ve cnjectue 1 by shwing that it imlies a lngest cycle dminates . Nevetheless, we exect
that the cnjectue hlds, and these examles simly shw that u lngest cycle techniques cannt genealize .
Acknwledgements
Reseach f the secnd auth is suted by NSERC Canada unde gant numbe
A0579 . Thanks t Adian Bndy f cmments which imved the esentatin .
Refeences
[1]
J .A . Bndy and U .S .R . Muty, Gah They with Alicatins, Macmillan, Lndn, Elsevie, New Yk, 1976 .
1 95
1W
0
[2]
0
Y2
Figue 5 .
J .A . Bndy, Lngest aths and cycles in gahs f high degee, Univesity f Watel Peint CORR 80-16 .
[3]
C .J . Clbun, J .M . Keil, and L.K . Stewat, Finding minimum dminating cycles in
emutatin gahs,
[4]
binatia ,
[5]
Oe . Res . Lettes , t aea .
C .J . Clbun, L .K . Stewat, Dminating cycles in seies-aallel gahs,
As Cm-
t aea .
G .A . Diac, Sme theems n abstact gahs,
Pc . Lndn Math . Sc., (3) 2
(1952), 69-81 .
[6]
P . Edös and T . Gallai, On maximal aths and cicuits f gahs,
Sci. Hunga . 10
[7]
Acta Math . Acad .
(1959) 337-356 .
A . Pskuwski, Minimum dminating cycles in 2-tees,
Int . J. Cm . Inf. Sci . ,
8 (1979) 405-417 .
[8]
H .J . Veldman, Existence f dminating cycles and aths,
Discete Mathematics ,
43 (1983) 281-296 .
Nte added (July 1985) :
Bndy and Fan (Univesity f Watel), and Faisse (Univesite de Pais Sud)
ecently cmmunicated t us diffeent fs f cnjectue 1 . We summaize Faisse's
f hee . Veldman [9] defines a D a cycle t be a cycle C f which G-C cntains n
cnnected cmnent with X me vetices . Futhe define as t be the maximum
numbe f emte subgahs f de X (tw subgahs ae emte if n edge cnnects a
1 96
vetex in ne t a vetex in the the) . Veldman [9, thm 2] ves that if a s < c, then
G is D a
cyclic when c>2 .
In geneal, a
DX
cycle need nt be dminating, but if ,\ <_ 6+1 such a cycle is dm-
inating . Nw set X = c+1
and set 6 > X+c . Cmute ax .
theem assues us that thee is a
Da
If as
< c,
then Veldman's
cycle, which (since X<6) is dminating . If n the
the hand, a s > c, thee ae at least c+1 emte subgahs each with X vetices . All
ae disjint, and this accunts f n-c vetices in ttal . The gah is c-cnnected, and
hence the emaining c vetices ae cnnected t each f the emte subgahs . Hweve,
cnside a vetex in ne f the emte subgahs . It has at mst X-1 neighbus in the
emte subgah, n neighbus in any the emte subgah, and at mst c neighbus
amng the cnnecting vetices . But then its degee is smalle than 6, which is a cntadictin . This ves cnjectue 1 .
Bndy and Fan ve a me geneal theem which has this esult as a cllay .
[9]
H .J . Veldman, "Existence f
D a -cycles
(1983) 309-316 .
1 97
and
D a-aths", Discete Mathematics 44