Hamiltonian Virus-Free Digraphs
Digrafos Libres de Virus Hamiltonianos
OsarOrdazandLeidaGonzalez(*)
(oordazanubis.iens.uv.ve)
DepartamentodeMatematiaand
CentrodeIngenieradeSoftwareYSistemasISYS.
FaultaddeCienias. UniversidadCentral deVenezuela
Ap. 47567,Caraas1041-A,Venezuela.
IsabelMarquez
UniversidadCentroidentalLisandroAlvarado
DeanatodeCienias. DepartamentodeMatematia.
Barquisimeto,Venezuela.
DomingoQuiroz(dquirozusb.ve)
UniversidadSimonBolivar. DepartamentodeMatematia.
Ap. 89000,Caraas1080-A,Venezuela.
Abstrat
A hamiltonian virus is a loal onguration that, if present in a
digraph, forbids thisdigraph to have a hamiltonianiruit.
Unfortu-nately,therearenon-hamiltoniandigraphsthatarehamiltonian
virus-free. Somefamiliesofthesedigraphswillbedesribedhere. Moreover,
problemsandonjeturesrelatedtohamiltonianvirus-freedigraphsare
given.
Keywordsand phrases: digraph,hamiltoniandigraph,hamiltonian
virus.
Resumen
Un virus hamiltoniano es una estrutura loal que, estando
pre-senteen un digrafo, impideque este tenga un iruito hamiltoniano.
Desafortunadamente, existen digrafos no hamiltonianos sin virus
ha-miltonianos. Algunasfamiliasdeestosdigrafossondesritasaqu. Mas
aun, seplanteanproblemasy onjeturas relativas adigrafos sin virus
hamiltonianos.
Palabras y frases lave: digrafos,digrafoshamiltonianos, virus
ha-miltonianos.
Reeived: 1999/06/04.Revised: 1999/10/28. Aepted: 1999/11/15.
1 Introdution and terminology
Theimportane ofhamiltonianviruses[1,8℄istheirrelationwiththe
\erti-ation" ofnon-hamiltonian digraph families, i.e. they arenon-hamiltonian
ifandonlyiftheyhaveahamiltonianvirus. Forexample,balanedbipartite
digraphsarehamiltonianifand onlyiftheyarehamiltonianvirus-free. This
paperhasmultiple goals:
To identify non-hamiltonian digraph families whih are hamiltonian
virus-free,anddigraphfamiliesthat arehamiltonianifandonlyifthey
arehamiltonianvirus-free.
To haraterizedigraphs that donot ontainhamiltonian virusesof a
givenorder.
To present and disuss problems and onjetures related to expeted
propertiesofhamiltonianvirus-freedigraphfamilies.
1.1 Terminology
The terminologydesribedin what followsis taken textuallyfrom [2℄and it
will beusedthroughoutthewholepaper.
Invariants areintegerorbooleanvaluesthat arepreservedunder
isomor-phism. Wewillbeusingthefollowinginvariants,relationsbetweeninvariants,
theorems anddigraphexamples. LetD=(V(D);E(D))beadigraph.
Integer invariants
nodes: numberofnodesofadigraph.
ars: numberofarsofadigraph.
alpha2: maximumsizeofasetofnodeswhihinduesnoiruitoflength2.
alpha0: maximumsizeofasetofnodesinduingnoar.
woodall: min fd +
(x)+d (y) : (x;y) 2= E(D);x 6= yg (if alpha2 = 1, then
woodall =2nodes byonvention.)
minimum: minfmindegpositive,mindegnegativeg.
mindegpositive minfd +
(x):x2V(D)g.
mindegnegative minfd (x):x2V(D)g.
Boolean invariants
hamiltonian: thedigraphontainsahamiltonian iruit.
traeable: thedigraphontainsahamiltonianpath.
bipartite: its vertex set is partitioned into twosubsets X and Y suh that
eaharhasonevertexin X andanotherin Y.
antisymmetri: itdoesnotontainairuitoflengthtwo.
(1,1)-fator: itontainsaspanningsubdigraphHsuhthatd +
H (x)=d
H (x)=
1forallverties.
Relationsbetween invariants
R
11
: minimum 2 ^ nodes4=)hamiltonian.
R
31
: antisymmetri=)arsnodes(nodes 1)/2.
Theorems and onjetures
Theorem 51: k-onneted^ (alpha0 k) =) (1,1)-fator. Best result: see
D
20 .
Theorem64: antisymmetri^(nodes2h+2)^(h2)^(minimumh)
=)hamiltonian.
Theorem 65: antisymmetri ^ (nodes 6) ^ (woodall nodes 2) =)
hamiltonian.
Theorem66: antisymmetri^(h5)^(minimumh)^(nodes2h+5)
=)hamiltonian.
Theorem67: antisymmetri^2-onneted^[arsnodes(nodes 1)/2 2℄
=)hamiltonian. Best result: seeD
20 :
Theorem 77: (nodes = 2a+1) ^ (minimum a) =) hamiltonian _D
5 _
D
6 _D
7 _D
8 .
Theorem 78: r-diregular^(nodes=2r+1)=)hamiltonian _D
5 _D
6 .
2 Hamiltonian viruses
A hamiltonian virus is a loal onguration that, if present in a digraph,
forbidsthisdigraphtohaveahamiltonianiruit.
Theorem1 ([1℄). Let H = (V(H);E(H)) be a proper indued subdigraph
of a given digraph D = (V(D);E(D)). A 3-uple (H;T +
;T ), where T +
=
fx 2 V(H) : d +
H
(x) = d +
D
(x)g and T = fx 2 V(H) : d
H
(x) = d
D (x)g,
is a hamiltonian virus if and only if for every set of disjoint direted paths
P
1 ;:::;P
r
overing V(H)there existsapath P
j =x
1
j :::x
q(j)
j
, withq(j)1
suh that either x 1
j
2 T or x q(j)
j 2 T
+
. The order of a hamiltonian virus
Inwhatfollowsa3-uple(H;T +
;T )ispresentinadigraphDifandonly
if there exists in D a proper indued subdigraph H 1
isomorphi to H (for
onveniene weidentify H 1
is a hamiltonian virusof a given digraphD; then wemust haveT +
agivendigraph, thenthere existsadigraphD where (H;T +
;T )is present
andD ishamiltonian.
Theorem2. If adigraphofordernisfreeof hamiltonianvirusesoforderh
for someh with 2h<n; then it has no hamiltonian viruses of order less
thanh.
Equivalently,fromahamiltonianvirusoforder1hn 2weanbuild
ahamiltonianvirusoforderh+1:
Proof. Let us reason ab absurdo. Let D be a digraph of order n and 2
h n 1: Assuming D ontains ahamiltonian virus (H;T +
;T ) of order
h 1. Letx2V(D)nV(H)andH
1
thesubdigraphinduedbyV(H[fxg)
in D. It islear thatthe 3-uple(H
is presentin D. Now,weshall seethat (H
1
)isahamiltonianvirus.
Let P
1 ;:::;P
r
be a set of disjointdireted paths overingV(H
1
): Without
lossofgeneralityweansupposethat x2V(P
1
):Weonsider threeases:
Case1P
isahamiltonianvirusthereexistsapathP 1
:Thereforex 1
. Thissituationwillbetreatedas
Case1.
. Therefore (H
1
) is a hamiltonian
virus oforder h. Aontradition.
Corollary 1. Ifadigraphofordernhashamiltonianvirusesthenitontains
FromCorollary1wehave:
(x)gisnotahamiltonianvirus thenD is
hamiltonianvirus-free. Moreover,ifD xisnotahamiltonianvirusforsome
x,thenforeahhamiltonianvirus(H;T +
;T )ofD wehavex2V(H):
DigraphD
6
showsthatthereexistnon-hamiltoniandigraphshamiltonian
virus-free. In D
6
hamiltonian viruses of order 6 are not present. Then by
Corollary1D
6
doesnothaveviruses.
FromTheorem1wehave:
Remark 2. Ahamiltonianvirus-freedigraphD hasthefollowingstruture:
for eahvertexxtheremainingpartD x hasaoveringbyvertexdisjoint
pathsP
1 ;:::;P
r
suhthateahoneof themmakesairuitwithx.
InthenextlemmaD[S℄denotesthesubdigraphinduedbyS.
Lemma1. Let (H;T
;T )isahamiltonian viruspresentin D.
Proof. LetP
1 ;:::;P
r
beasetof disjointdireted pathsoveringV(H S):
Thenforanysetofdisjointdireted pathsP
r+1 ;:::;P
s
overingV(D[S℄)we
havethatP
1
isasetofdisjointdiretedpathsovering
V(H). Sine(H;T +
;T )is ahamiltonianvirus, there exists apath P
j
Theorem3. Let D be an antisymmetridigraph with minimum 2. Then
D isfreeof hamiltonian virusesoforder 4.
Proof. Let us reason ab absurdo. Let (H;T +
j3(similarly,jT j3). Byasimpleinspetionontherelative
positionsofthearsinH,weandeduethateitherthereexistsasymmetri
ar or(H;T +
;T )isnotahamiltonianvirus. Aontradition. Thereforeit
mustbe jT +
j 1 and jT j 1. If T +
= ;then T 6= ;, henethere are
disjoint direted paths overingV(H) that do not verify the onditions for
(H;T +
;T ) to bea hamiltonian virus. A ontradition. Consider now the
treated). Byenumeratingseveral ases resultingfrom the relative positions
of thearsin H,sets of disjointdireted pathsoveringV(H)donotverify
the onditionin order that(H;T +
;T )is ahamiltonian virus. Forthease
jT +
j =jT j =1( orjT +
j =jT j =2), wean seethe existene ofsets of
disjointdireted pathsoveringV(H)thatdonotverifytheonditionstobe
ahamiltonianvirus. ByTheorem2,Dis ahamiltonianvirus-freedigraphof
order 3.
InTheorem4,weneedthefollowingdenition:
Denition 1. [1℄A1-onnetedvirusisaloalongurationthat,ifpresent
in adigraph,forbidsthisdigraphtobe1-onneted. LetH =(V(H);E(H))
beaproperinduedsubdigraph ofagivendigraphD=(V(D);E(D)): A
3-uple (H;T +
;T ); whereT +
=fx2V(H):d +
H (x)=d
+
D
(x)gandT =fx2
V(H):d
H (x)=d
D
(x)g,isa1-onnetedvirus ifandonlyifV(H)=T +
or
V(H)=T .
Theorem4. A hamiltonian virus-freedigraphis2-onneted.
Proof. Let D = (V(D);E(D)) be hamiltonian virus-free. Letus reason ab
absurdo. Let (H;T +
;T ) bea1-onneted virus presentin D x forsome
x2V(D)withV(H)=T :TheaseV(H)=T +
istreatedinasimilarway.
Let y 2 V(D x)nV(H): By Remark 2, there exist vertex disjoint paths
P
1 ;:::;P
r
overingD y suh that eah oneof them makesairuit with
y. Sine V(H) = T then for eah P
j = x
1
j :::x
q(j)
j
we have x 1
j = 2 V(H):
Moreover, for eah x t
i
2 V(H)\V(P
i
)wehave x t 1
i
2 V(H). Hene x 1
i 2
V(H):Aontradition.
3 Hamiltonian virus-free digraph families
In this setion we desribe non-hamiltonian and hamiltonian virus-free
di-graph families. There exist non-hamiltonian digraph families with
hamilto-nian viruses. This fat hasallowedto deriveproblems and onjeturesthat
arepresentedanddisussedin thissetion.
Theorem5. Balaned bipartite digraphs are hamiltonian ifandonly if they
arehamiltonian virus-free.
Proof. LetD =(X[Y;E(D))beahamiltonianbalaned bipartitedigraph.
Remark2,D xhasaoveringbyvertexdisjointpathsP
1 ;:::;P
r
suhthat
eahoneofthemmakesairuitwithx. LetC
i
(1ir)betheseiruits.
Sine D isabalaned bipartitedigraph wehavejV(C
i
)j=2n
i
(the iruits
haveevenlength),jXj=n
1 +n
2
1++n
r
1andjYj=n
1 +n
2
++n
r .
ThereforeD isnotbalaned. Aontradition.
Thenextremarkfollowsdiretly fromTheorem77.
Remark 3. There are no non-hamiltonian and hamiltonian virus-free
di-graphs with minimum = 2 and nodes = 5. Notie that digraph D
5 has
hamiltonianvirus.
Theonlynon-hamiltonianand hamiltonianvirus-freedigraphwith
mini-mum=3and nodes=7isD
6 .
A non-hamiltoniandigraph withnodes =2minimum+19has
hamil-tonian viruses. By Theorem 77 the only families of digraphs that are
non-hamiltonian and where nodes =2minimum+19 holds, are D
7 and D
8 .
These familieshaveviruses.
Proposition1. A hamiltonianvirus-freedigraph withnodes5is
hamilto-nian.
Proof. Let D be a hamiltonian virus-free digraph; then minimum 2. If
nodes 4 then, by R
11
, D is hamiltonian. The ase nodes = 5 follows
diretlyfromRemark3.
Proposition2. Ahamiltonianvirus-freedigraphwithminimum=2is
trae-able orhamiltonian.
Proof. Sine minimum =2, there exists x 2 V(D) suh that d +
(x) = 2or
d (x)=2. ThenbyRemark 2,thereexist atmosttwovertexdisjointpaths
overing D x, say P
i = x
1
i x
2
i :::x
r(i)
i
(1 i 2), suh that eah one of
themmakesairuitwithx. IfthereisonlyonepaththenD ishamiltonian,
otherwisethepathx 1
1 x
2
1 :::x
r(1)
1 xx
1
2 x
2
2 :::x
r(2)
2
makesD traeable.
Proposition3. A hamiltonian virus-free antisymmetri digraph with nodes
=6,7or8ishamiltonian ortraeable. Moreover, the onlyhamiltonian
non-hamiltonian virus-freeantisymmetridigraphwith nodes=7isdigraph EX.
Proof. Let D beahamiltonian virus-freedigraph. Then minimum 2 and
woodall 4. By Theorem 65, if nodes = 6 then D is hamiltonian. For
Thefollowingonjetureshould betrue:
Conjeture 1. A hamiltonian virus-free antisymmetri digraph is
hamilto-nian ortraeable.
NotiethatdigraphEX isnon-hamiltonian, butitistraeable.
Thefollowingonjetureshouldbetrue:
Conjeture 2. Ahamiltonianvirus-freeantisymmetrir diregulardigraph
with r3andnodes4r+1ishamiltonian.
We an formulate the following remarks for Conjeture 2: ByTheorem
64,theonjetureforr=3istruewhennodes8. Forase9nodes13
the hypothesis hamiltonianvirus-free perhapsanbeuseful. Notie that by
Theorem78,theonjetureistruefornodes=2r+1. Theonjetureistrue
from Theorem66forr=5andnodes15.
Problem 1. LetD bean antisymmetriandhamiltonian virus-freedigraph.
Findthegreatestpositiveintegerx suhthatwhenarsnodes(nodes-1)/2 x
then D ishamiltonian.
ByTheorem4andTheorem67wehavex2. MoreoverthedigraphD
20
showsthatTheorem67isthebest possible. NotiethatD
20
hashamiltonian
viruses.
Problem 2. LetDbeak-onnetedandhamiltonianvirus-freedigraph. Find
the greatestintegerxsuhthatwhenalpha0k+x thenD ontainsa
(1,1)-fator.
ByTheorem 51wehave that x 0. Moreoverdigraph D
20
showsthat
this theoremisthebestpossible. Notiethat D
20
hashamiltonianviruses.
3.1 Hamiltonian virus-free hypohamiltonian digraphs
This setion is devoted to study hypohamiltonianhamiltonian virus-free
di-graphsand those that havehamiltonian viruses. Themethods, forbuilding
hypohamiltonian digraphs, established in [9℄ and [4℄ are given. Some
on-jetures related to hamiltonianvirus-free and hypohamiltoniandigraphs are
disussed.
AdigraphDishypohamiltonianifithasnohamiltonianiruitsbutevery
Conjeture 3. Everyhamiltonian virus-freenon-hamiltoniandigraphis
hy-pohamiltonian.
Ortheweakerone:
Conjeture 4. Every non-hamiltonian vertex-transitive hamiltonian
virus-freedigraph ishypohamiltonian.
Conjeture 5. Everyhypohamiltonian digraphishamiltonian virus-free.
Notie that digraph D
6
is hypohamiltonian and hamiltonian virus-free.
In[9℄Thomassengivesamethod forobtaininghypohamiltoniandigraphsby
formingtheartesianprodutofyles. Wegivehereashortsummaryofhis
results,in ordertogivesomeremarksonConjetures3,4,5.
ReallthatifD
1 andD
2
aredigraphsthenitsartesianprodutD
1 D
2
isthedigraphwithvertexsetV(D
1
this notationThomassen givesthefollowingtheorems:
Theorem6 ([9℄). Foreahk3;m2;C
k C
mk 1
isahypohamiltonian
antisymmetri digraph. Moreover, C
3 C
6k +4
is hypohamiltonian for eah
k0.
Theorem7 ([9℄). There isno hypohamiltonian digraph with fewer thansix
verties, andfor eah odd m3,C
2 C
m
is ahypohamiltonian digraph.
Remark 4. ThehypohamiltoniandigraphsC
3 C
6k +4
withk0(Theorem
6)andthehypohamiltoniandigraphsgiveninTheorem7verifyConjeture5.
Howeverthedigraph C
4 C
11
, i.e., k=4and m=3in Theorem 6,refutes
Conjeture5. Wehaveprovedthattheonlynon-hamiltonianvertex-transitive
digraph whih is also hamiltonian virus-freeof order 6, is the
hypohamilto-nian digraph C
2 C
3
: Whih is in favorof Conjeture 4. Nevertheless the
Conjeture 4is false, the digraphEX isnon-hamiltonian, vertex-transitive,
hamiltonianvirus-freeandnothypohamiltonian.
In[4℄Fouquet andJolivetgivethefollowingtheoremforobtaining
hypo-hamiltoniandigraphs.
sribedbelowishypohamiltonian.
Forn=2p+1andp3: V(F
takenmodulo 2p:
For n = 2p and p 4, F
n
is obtained from F
2p 1
replaing the ar
x
andaddingthefollowingars: x
2p 4
:Eah index istakenmodulo 2p 2:
InthenexttheoremletC =x
beairuit. Wedenote by
C(x
i ;x
j
)theinduedpathofC beginningat x
i
andendingatx
j .
Theorem9. Foreah n8;F
n
ishamiltonian virus-free.
Proof. WefollowRemark1andRemark2. Ineahstepoftheproof,weshow
the pathsP
i
). Weonsider twoases:
Case 1n=2p+1andp4:
Remark 5. For n8,thehypohamiltoniandigraphsF
n
4 Conlusion
It is well known that the problem to deide when a digraphis hamiltonian
is NP-omplete[3℄. A \yes"answertothehamiltoniity problemforagiven
digraph an be veried by heking in polynomial time that a sequene of
vertiesgivenbyanoraleisahamiltonianiruit.Inaseofnon-hamiltonian
digraphs, asstatedin [7℄ pages28, 29, there isno known wayof verifyinga
\yes"answertotheomplementary problemof deidingifadigraphis
non-hamiltonian. A solutionto this problem is to provide a hamiltonian virus,
whose presene in the digraph an also be heked in polynomial time. In
aseof thenon-hamiltonianhamiltonian virus-freedigraphs,theymusthold
the partiularstruture givenin Remark 2. Thevirus notionhasbeenused
in randomgenerationofdigraphswithoutertainproperties[8℄.
Wehavebuilt aninterativesupport toolalled GRAPHVIRUS [5℄that
allowsthegraphial edition ofhamiltonian viruses and theveriationthat
agivenstrutureisahamiltonianvirus. GRAPHVIRUSanalsobeusedto
deriveaproedure fordeidingwhether agiven digraphisnon-hamiltonian.
Thisproedureisofthesameomplexityoftheproblemofdeidingifagiven
digraph ishamiltonian, butthe interestoftheproedure isthefat ofusing
aloalstruture.
Finally,thetheoretiinterestoftheresultspresentedhereistheirrelation
with the extension of known suÆient onditionswith the new hamiltonian
virus-freeonditionfortheexisteneofhamiltonianiruits.
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