Directory UMM :Journals:Journal_of_mathematics:DM:
Divulga
iones Matemati
as Vol. 8 No. 1 (2000), pp. 1{13
Hamiltonian Virus-Free Digraphs
Digrafos Libres de Virus Hamiltonianos
Os
ar Ordaz and Leida Gonz
alez (*)
oordazanubis.
iens.u
v.ve)
(
Departamento de Matem
ati
a and
Centro de Ingenier
a de Software Y Sistemas ISYS.
Fa
ultad de Cien
ias. Universidad Central de Venezuela
Ap. 47567, Cara
as 1041-A, Venezuela.
Isabel M
arquez
Universidad Centro
idental Lisandro Alvarado
De
anato de Cien
ias. Departamento de Matem
ati
a.
Barquisimeto, Venezuela.
Domingo Quiroz (
dquirozusb.ve)
Universidad Sim
on Bolivar. Departamento de Matem
ati
a.
Ap. 89000, Cara
as 1080-A, Venezuela.
Abstra
t
A hamiltonian virus is a lo
al
on guration that, if present in a
digraph, forbids this digraph to have a hamiltonian
ir
uit. Unfortunately, there are non-hamiltonian digraphs that are hamiltonian virusfree. Some families of these digraphs will be des
ribed here. Moreover,
problems and
onje
tures related to hamiltonian virus-free digraphs are
given.
Keywords and phrases: digraph, hamiltonian digraph, hamiltonian
virus.
Resumen
Un virus hamiltoniano es una estru
tura lo
al que, estando presente en un digrafo, impide que este tenga un
ir
uito hamiltoniano.
Desafortunadamente, existen digrafos no hamiltonianos sin virus hamiltonianos. Algunas familias de estos digrafos son des
ritas aqu. Mas
aun, se plantean problemas y
onjeturas relativas a digrafos sin virus
hamiltonianos.
Palabras y frases
lave: digrafos, digrafos hamiltonianos, virus hamiltonianos.
Re
eived: 1999/06/04. Revised: 1999/10/28. A
epted: 1999/11/15.
MSC (1991): 05C45, 05C40.
(*) Resear
h partially supported by
CDCH proje
t. No 03-12-3726-98.
GraphVirus
2
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
1
Introdu
tion and terminology
The importan
e of hamiltonian viruses [1, 8℄ is their relation with the \
erti
ation" of non-hamiltonian digraph families, i.e. they are non-hamiltonian
if and only if they have a hamiltonian virus. For example, balan
ed bipartite
digraphs are hamiltonian if and only if they are hamiltonian virus-free. This
paper has multiple goals:
To identify non-hamiltonian digraph families whi
h are hamiltonian
virus-free, and digraph families that are hamiltonian if and only if they
are hamiltonian virus-free.
To
hara
terize digraphs that do not
ontain hamiltonian viruses of a
given order.
To present and dis
uss problems and
onje
tures related to expe
ted
properties of hamiltonian virus-free digraph families.
1.1
Terminology
The terminology des
ribed in what follows is taken textually from [2℄ and it
will be used throughout the whole paper.
Invariants are integer or boolean values that are preserved under isomorphism. We will be using the following invariants, relations between invariants,
theorems and digraph examples. Let D = (V (D); E (D)) be a digraph.
Integer invariants
nodes : number of nodes of a digraph.
ar
s : number of ar
s of a digraph.
alpha2 : maximum size of a set of nodes whi
h indu
es no
ir
uit of length 2.
alpha0 : maximum size of a set of nodes indu
ing no ar
.
woodall : minfd+ (x) + d (y ) : (x; y ) 2
= E (D ); x 6= y g (if alpha2 = 1, then
woodall = 2nodes by
onvention.)
minimum : minfmindegpositive, mindegnegativeg.
mindegpositive minfd+ (x) : x 2 V (D)g.
mindegnegative minfd (x) : x 2 V (D)g.
Boolean invariants
hamiltonian : the digraph
ontains a hamiltonian
ir
uit.
tra
eable : the digraph
ontains a hamiltonian path.
k -
onne
ted : the deletion of fewer than k verti
es always results in a
onne
ted
digraph.
3
Hamiltonian virus-free digraphs
bipartite : its vertex set is partitioned into two subsets
ea
h ar
has one vertex in
X
and another in
Y
and
X
Y
su
h that
.
antisymmetri
: it does not
ontain a
ir
uit of length two.
(1,1)-fa
tor : it
ontains a spanning subdigraph H su
h that
d
1 for all verti
es.
+ (x) = d
H
H (x) =
Relations between invariants
11 :
R31 :
R
2 ^ nodes 4 =) hamiltonian.
antisymmetri
=) ar
s nodes (nodes
1)/2.
minimum
Theorems and
onje
tures
Theorem 51:
D20 .
k -
onne
ted
^ (alpha0
Theorem 64: antisymmetri
=
hamiltonian.
)
)
h+
^
(nodes
6_
D
7_
D
8.
Theorem 78: r-diregular
Digraphs examples
^
6)
^
h
(woodall
r
h)
nodes
a)
)
h+
1)/2
) hamiltonian _
=
D
) hamiltonian _ 5 _
+ 1) =
D
h)
2) =
^ (nodes 2
^ [ar
s nodes(nodes
^ (minimum
^ (nodes= 2
Best result: see
^ ( 2) ^ (minimum
h
Theorem 77: (nodes = 2a + 1)
D
2)
^ ( 5) ^ (minimum
Theorem 67: antisymmetri
2-
onne
ted
=
hamiltonian. Best result: see D20 :
)
) (1,1)-fa
tor.
=
^ (nodes 2
Theorem 65: antisymmetri
hamiltonian.
Theorem 66: antisymmetri
=
hamiltonian.
k)
D
6.
5)
2℄
5_
4
2
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
Hamiltonian viruses
A hamiltonian virus is a lo
al
on guration that, if present in a digraph,
forbids this digraph to have a hamiltonian
ir
uit.
Let H = (V (H ); E (H )) be a proper indu
ed 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 ) : dH (x) = dD (x)g,
is a hamiltonian virus if and only if for every set of disjoint dire
ted paths
q(j)
1
P1 ; : : : ; Pr
overing V (H ) there exists a path Pj = xj : : : x
j , with q(j ) 1
su
h that either x1j 2 T or xjq(j ) 2 T + . The order of a hamiltonian virus
(H; T + ; T ) is de ned as the
ardinality of V (H ).
Theorem 1 ([1℄).
5
Hamiltonian virus-free digraphs
In what follows a 3-uple (H; T + ; T
if there exists in
onvenien
e we identify
D (x)g
+
and
d
=
T
) is present in a digraph
a proper indu
ed subdigraph
D
f 2
H
x
1
V
with
(H ) :
H ),
d
su
h that
H (x)
=
is a hamiltonian virus of a given digraph
6= ;.
T
d
a given digraph, then there exists a digraph
D
+
D (x)g.
D;
In [1℄ we show that if a 3-uple (H; T + ; T
and
T
H
1
D
if and only
isomorphi
to
f 2
=
x
V
(H ) :
+
H+(x) =
Moreover if (H; T
then we must have
T
(for
H
d
+
;T
)
6= ; or
) is not a hamiltonian virus for
D
where (H; T + ; T
) is present
is hamiltonian.
Theorem 2. If a digraph of order n is free of hamiltonian viruses of order h
2
for some h with
h < n; then it has no hamiltonian viruses of order less
than h.
Equivalently, from a hamiltonian virus of order 1
a hamiltonian virus of order
Proof.
h
h
h
+ 1:
Let us reason ab absurdo. Let
1: Assuming
n
1. Let
x
2
V
(D )
n
D
V
D
h
2 we
an build
n
be a digraph of order
n
ontains a hamiltonian virus (H; T + ; T
(H ) and
H1
the subdigraph indu
ed by
+
+
+
D.
It is
lear that the 3-uple (H1 ; T1
r
loss of generality we
an suppose that
P
i
s
x
; T1
2
V
2
) with
i
r)
T
V
(H
[ f g)
x
T
(P1 ): We
onsider three
ases:
1
1
Case 1 P1 = xx1 : : : x1 : In this
ase x1 = T
(2
T1
and
1 = T
+
is present in D . Now, we shall see that (H1 ; T1 ; T1 ) is a hamiltonian virus.
Let P1 ; : : : ; P be a set of disjoint dire
ted paths
overing V (H1 ): Without
in
=
and 2
) of order
. Hen
e
are disjoint dire
ted paths
overing
V
1
P1
=
s
1
i1 =
x1 : : : x1 , P
(H ). Sin
e (H; T
+
;T
)
1
=
q(j) su
h that x1 2 T
is a hamiltonian virus there exists a path Pj = xj : : : xj
j
q
(
j
)
q
(j )
+
+
1
2 T : Therefore xj 2 T1 or xj 2 T1 :
or xj
s = T +. This situation will be treated as
s
1
Case 2 P1 = x1 : : : x1 x. Hen
e x1 2
1
Case 1.
Case 3 P
i+1 : : : xs , P 11
x1
1
i+1
+
=
1
i
x1 : : : x1 xx
i+1 : : : xs .
1
1
1
In this
ase
1
P1
=
1
i
x1 : : : x1 , P2
i (2 i r) are disjoint dire
ted paths
overing V (H ).
Sin
e x 2
= T
[ T and (H; T + ; T ) is a hamiltonian virus, x11 2 T or xs1 2
+
1
= T
there exists a path Pj1 (3 j r + 1) su
h
= T
and x1
T . In
ase x1 2
1 2
q(j 1)
+
that x1
j 1 2 T or xj 1 2 T +. Therefore (H1 ; T1 ; T1 ) is a hamiltonian
virus of order
h.
=
P
A
ontradi
tion.
Corollary 1. If a digraph of order n has hamiltonian viruses then it
ontains
a hamiltonian virus of order n
Proof.
Dire
tly from Theorem 2.
1.
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
6
From Corollary 1 we have:
Remark 1. Let D = (V (D ); E (D )) be a digraph. If for ea
h x
+
+
+
3-uple (H; T
and
f 2
=
T
) with
;T
x
(H ) :
V
d
=
D
hamiltonian virus-free. Moreover, if
x,
=
x; T
D
f 2
Digraph
Corollary 1
x
V
(H ) :
V
H (x)
(D ); the
=
d
+
d
is not a hamiltonian virus for some
x
then for ea
h hamiltonian virus (H; T + ; T
) of
D
we have
x
2
V
(H ):
shows that there exist non-hamiltonian digraphs hamiltonian
D6
virus-free. In
2
D (x)g
H (x) = dD (x)g is not a hamiltonian virus then D is
H
hamiltonian viruses of order 6 are not present. Then by
D6
does not have viruses.
D6
From Theorem 1 we have:
Remark 2. A hamiltonian virus-free digraph D has the following stru
ture:
for ea
h vertex
paths
x
r
the remaining part
In the next lemma
Lemma 1. Let (H; T
+
V
n(
(H )
Then
[
T
(H
Let
Proof.
D [S ℄
+
6 ;
)=
T
S; T
D
x
+
r
)
have that
s)
2
D T+
DT
r r
r
j
1
T
x
j
or x
r:
q(j)
j 2 T + : On the other hand sin
e S \ (T + [ T
Therefore (H
Let us reason ab absurdo.
of order 4 present in
S; T
+
;T
D.
4
;,
2
. Then
.
Let (H; T + ; T
Let us see that 1
)=
) is a hamiltonian virus.
) be a hamiltonian virus
j j 2 and 1 j j 2.
T
+
T
j j 3 (similarly, j j 3). By a simple inspe
tion on the relative
T
+
T
positions of the ar
s in
ar
or (H; T + ; T
H,
we
an dedu
e that either there exists a symmetri
) is not a hamiltonian virus. A
ontradi
tion. Therefore it
j j 1 and j j 1.
T
+
T
disjoint dire
ted paths
overing
(H; T + ; T
ase
S ):
s is a set of disjoint dire
ted paths
overing
Pj (1
D is free of hamiltonian viruses of order
must be
(H
) is a hamiltonian virus, there exists a path
Theorem 3. Let D be an antisymmetri
digraph with minimum
Suppose
V
s
overing V (D[S ℄) we
P +1 ; : : : ; P
P1 ; : : : ; P ; P +1 ; : : : ; P
with
Proof.
0.
is a hamiltonian virus present in D .
(H ). Sin
e (H; T + ; T
2
it must be 1
j
S.
be a set of disjoint dire
ted paths
overing
P1 ; : : : ; P
x.
be a hamiltonian virus present in D . Let S
+
S, d
(x) = d [ ℄ (x) =
[
℄
be su
h that for all x
)
;T
;T
denotes the subdigraph indu
ed by
Then for any set of disjoint dire
ted paths
V
has a
overing by vertex disjoint
su
h that ea
h one of them makes a
ir
uit with
P1 ; : : : ; P
V
If
T
+
=
; then
T
6= ;, hen
e there are
(H ) that do not verify the
onditions for
) to be a hamiltonian virus. A
ontradi
tion. Consider now the
j j = 1 and j j = 2 (the
ase j j = 2 and j j = 1 is similarly
T
+
T
T
+
T
Hamiltonian virus-free digraphs
7
treated). By enumerating several
ases resulting from the relative positions
of the ar
s in H , sets of disjoint dire
ted paths
overing V (H ) do not verify
the
ondition in order that (H; T + ; T ) is a hamiltonian virus. For the
ase
jT +j = jT j = 1 ( or jT +j = jT j = 2), we
an see the existen
e of sets of
disjoint dire
ted paths
overing V (H ) that do not verify the
onditions to be
a hamiltonian virus. By Theorem 2, D is a hamiltonian virus-free digraph of
order 3.
In Theorem 4, we need the following de nition:
De nition 1. [1℄ A 1-
onne
ted virus is a lo
al
on guration that, if present
in a digraph, forbids this digraph to be 1-
onne
ted. Let H = (V (H ); E (H ))
be a proper indu
ed 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) = dD (x)g and T = f+x 2
V (H ) : d (x) = d (x)g, is a 1-
onne
ted virus if and only if V (H ) = T
or
H
D
V (H ) = T .
Theorem 4.
A hamiltonian virus-free digraph is 2-
onne
ted.
Let D = (V (D); E (D)) be hamiltonian virus-free. Let us reason ab
absurdo. Let (H; T + ; T ) be a 1-
onne
ted virus present in D x for some
+
x 2 V (D ) with V (H ) = T : The
ase V (H ) = T
is treated in a similar way.
Let y 2 V (D x) n V (H ): By Remark 2, there exist vertex disjoint paths
P1 ; : : : ; Pr
overing D
y su
h that ea
h one of them makes a
ir
uit with
y . Sin
e V (H ) = T
then for ea
h Pj = x1j : : : xjq(j ) we have x1j 2= V (H ):
Moreover, for ea
h xti 2 V (H ) \ V (Pi ) we have xti 1 2 V (H ). Hen
e x1i 2
V (H ): A
ontradi
tion.
Proof.
3
Hamiltonian virus-free digraph families
In this se
tion we des
ribe non-hamiltonian and hamiltonian virus-free digraph families. There exist non-hamiltonian digraph families with hamiltonian viruses. This fa
t has allowed to derive problems and
onje
tures that
are presented and dis
ussed in this se
tion.
Theorem 5.
Balan
ed bipartite digraphs are hamiltonian if and only if they
are hamiltonian virus-free.
Let D = (X [ Y ; E (D)) be a hamiltonian balan
ed bipartite digraph.
Hen
e D is hamiltonian virus-free (by Theorem 1 ). Let us assume that D
is hamiltonian virus-free. Let x 2 X (similarly dis
ussed for y 2 Y ). By
Proof.
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
8
Remark 2,
D
has a
overing by vertex disjoint paths
x
P1 ; : : : ; Pr
su
h that
) be these
ir
uits.
Sin
e
is a balan
ed bipartite digraph we have j ( )j = 2
(the
ir
uits
have even length), j j =
+
1+ +
1 and j j =
+
+ + .
ea
h one of them makes a
ir
uit with
x.
Let
(1
Ci
D
i
V
X
Therefore
D
n1
n2
r
Ci
nr
ni
Y
n1
n2
nr
is not balan
ed. A
ontradi
tion.
The next remark follows dire
tly from Theorem 77.
Remark 3. There are no non-hamiltonian and hamiltonian virus-free di-
graphs with
= 2 and
minimum
nodes
= 5.
Noti
e that digraph
D5
has
hamiltonian virus.
The only non-hamiltonian and hamiltonian virus-free digraph with
= 3 and
mum
nodes
= 7 is
D6 .
A non-hamiltonian digraph with
= 2minimum + 1
nodes
mini-
9 has hamil-
tonian viruses. By Theorem 77 the only families of digraphs that are nonhamiltonian and where
nodes
= 2minimum + 1
These families have viruses.
9 holds, are
Proposition 1. A hamiltonian virus-free digraph with nodes
D7
5
and
D8 .
is hamilto-
nian.
Proof.
nodes
Let
D
be a hamiltonian virus-free digraph; then
4 then, by
R11 , D
is hamiltonian.
The
ase
minimum
2.
If
= 5 follows
nodes
dire
tly from Remark 3.
Proposition 2. A hamiltonian virus-free digraph with minimum = 2 is tra
eable or hamiltonian.
Proof.
d
Sin
e
minimum
= 2, there exists
x
2
V
(D ) su
h that
+
d
(x) = 2 or
(x) = 2. Then by Remark 2, there exist at most two vertex disjoint paths
overing
x,
D
say
Pi
=
them makes a
ir
uit with
otherwise the path
r (i)
1 2
(1
xi xi : : : xi
x.
2), su
h that ea
h one of
i
If there is only one path then
r (1)
r (2)
1 2
1 2
x1 x1 : : : x1
xx2 x2 : : : x2
makes
D
is hamiltonian,
D tra
eable.
Proposition 3. A hamiltonian virus-free antisymmetri
digraph with nodes
= 6,
7 or 8 is hamiltonian or tra
eable. Moreover, the only hamiltonian non-
hamiltonian virus-free antisymmetri
digraph with
Proof.
Let
woodall
nodes
D
4.
be a hamiltonian virus-free digraph. Then
By Theorem 65, if
= 7 or 8, if
tra
eable.
If
nodes = 7
minimum
minimum
nodes
= 6 then
D
= 2 then, by Proposition 2,
3 then, by Theorem 65,
D
is digraph E X .
minimum
2 and
is hamiltonian.
D
is
For
hamiltonian
is hamiltonian.
or
9
Hamiltonian virus-free digraphs
The following
onje
ture should be true:
Conje
ture 1. A hamiltonian virus-free antisymmetri
digraph is hamiltonian or tra
eable.
Noti
e that digraph EX is non-hamiltonian, but it is tra
eable.
The following
onje
ture should be true:
Conje
ture 2.
with
A hamiltonian virus-free antisymmetri
r 3 and nodes 4r + 1 is hamiltonian.
r diregular digraph
We
an formulate the following remarks for Conje
ture 2: By Theorem
64, the
onje
ture for r = 3 is true when nodes 8. For
ase 9 nodes 13
the hypothesis hamiltonian virus-free perhaps
an be useful. Noti
e that by
Theorem 78, the
onje
ture is true for nodes = 2r + 1. The
onje
ture is true
from Theorem 66 for r = 5 and nodes 15.
D
Problem 1. Let
be an antisymmetri
and hamiltonian virus-free digraph.
Find the greatest positive integer su
h that when ar
s nodes(nodes-1)/2
then
is hamiltonian.
D
x
x
By Theorem 4 and Theorem 67 we have x 2. Moreover the digraph D20
shows that Theorem 67 is the best possible. Noti
e that D20 has hamiltonian
viruses.
D k
x
Problem 2. Let
be a -
onne
ted and hamiltonian virus-free digraph. Find
the greatest integer su
h that when
then
ontains a (1,1)fa
tor.
alpha0 k + x
D
By Theorem 51 we have that x 0. Moreover digraph D20 shows that
this theorem is the best possible. Noti
e that D20 has hamiltonian viruses.
3.1
Hamiltonian virus-free hypohamiltonian digraphs
This se
tion is devoted to study hypohamiltonian hamiltonian virus-free digraphs and those that have hamiltonian viruses. The methods, for building
hypohamiltonian digraphs, established in [9℄ and [4℄ are given. Some
onje
tures related to hamiltonian virus-free and hypohamiltonian digraphs are
dis
ussed.
A digraph D is hypohamiltonian if it has no hamiltonian
ir
uits but every
vertex-deleted subdigraph D v has su
h a
ir
uit.
It is natural to formulate, as in [6℄, the following
onje
tures:
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
10
Conje
ture 3.
Every hamiltonian virus-free non-hamiltonian digraph is hy-
pohamiltonian.
Or the weaker one:
Conje
ture 4.
Every non-hamiltonian vertex-transitive hamiltonian virus-
free digraph is hypohamiltonian.
Conje
ture 5.
Every hypohamiltonian digraph is hamiltonian virus-free.
Noti
e that digraph
D
6
is hypohamiltonian and hamiltonian virus-free.
In [9℄ Thomassen gives a method for obtaining hypohamiltonian digraphs by
forming the
artesian produ
t of
y
les. We give here a short summary of his
results, in order to give some remarks on Conje
tures 3, 4, 5.
Re
all that if
D
1
and
D
2
is the digraph with vertex set
are digraphs then its
(D1 )
V
to (u1 ; u2 ) is present if and only if
1 1
v u
2
E (D
1 ):
v
V
artesian produ
t D
1 D2
(D2 ) su
h that the edge from (v1 ; v2 )
1 = u1 and v2 u2 2 E (D2 ), or v2 = u2 and
The dire
ted
y
le of length
k;
2
k;
is denoted
Ck .
With
this notation Thomassen gives the following theorems:
Theorem 6 ([9℄).
For ea
h k
antisymmetri
digraph.
k
0
3
2 k
3 6k+4
; m
Cmk
; C
Moreover, C
C
1
is a hypohamiltonian
is hypohamiltonian for ea
h
.
Theorem 7 ([9℄).
There is no hypohamiltonian digraph with fewer than six
verti
es, and for ea
h odd m
3
, C
2 Cm
is a hypohamiltonian digraph.
Remark 4. The hypohamiltonian digraphs C3
C
6k+4 with k 0 (Theorem
6) and the hypohamiltonian digraphs given in Theorem 7 verify Conje
ture 5.
However the digraph
C
4 C11 , i.e.,
= 4 and
k
m
= 3 in Theorem 6, refutes
Conje
ture 5. We have proved that the only non-hamiltonian vertex-transitive
digraph whi
h is also hamiltonian virus-free of order 6, is the hypohamiltonian digraph
C
2 C3 :
Whi
h is in favor of Conje
ture 4. Nevertheless the
Conje
ture 4 is false, the digraph
EX
is non-hamiltonian, vertex-transitive,
hamiltonian virus-free and not hypohamiltonian.
In [4℄ Fouquet and Jolivet give the following theorem for obtaining hypohamiltonian digraphs.
Theorem 8 ([4℄).
For ea
h n
s
ribed below is hypohamiltonian.
For n
= 6,
6 = C2 C3 :
F
6;
the digraph Fn
= (V (Fn ); E (Fn ))
de-
11
Hamiltonian virus-free digraphs
= 2p + 1
For n
f
2p
1; k
= 2p
For n
(0
g[ f
odd
=
2 1 ; yg and E (Fn ) =
+4 ; 1 k
xo ; x ; : : : ; x p
2)
p
xk y and yxk ;
and p
f 1
g[f
0 2
Fn )
: V
i
2p:
taken modulo
3 (
2
and p
2 1 xo ; xi xi+1
x p
xk xk
k
p
2
and xk xk
1; k
4
g
even : Ea
h index is
2 1
, Fn is obtained from F p
repla
ing the ar
2 3
2 3
2 4 x2p 3 ;
x
;
x
x;
x
x
;
x
x
;
x
x;
xx
;
x
x
2 3 2p 4 o
2p 3 2p 5 2p 7 2p 3 1
3 o 2p 4 ; x2p 3 x;
xx2p 3 : Ea
h index is taken modulo 2p
2:
In the next theorem let C = xo x1 : : : x2p 1 xo be a
ir
uit. We denote by
xo by the path x p
x p
xxo and adding the following ar
s: x p
x p
C (xi ; xj )
the indu
ed path of
Theorem 9. For ea
h n
Proof.
w
2
8
V
(1
Pi
2) that
over
i
Fn
xk
For
k
For
k
Fn
2
k
the paths are
For
xk
1 = C (x2 ; x3 )x1 x2p 1 xo
4
with 0 2
and
P
2 = C (x4 ; x2p 2 ):
:
k
3.
p
For
k
= 3 the paths are
k
P
P
odd ex
ept 3
Fn
w.
1.
p
k
Fn
and make a
ir
uit with
1 = y and P2 = C (xk+1 ; xk 1 ).
odd, the paths are P1 = C (xk+4 ; xk 4 ) and P2 = C (xk+1 ; xk+3 )y
3 ; xk 1 ).
and
2p
3 the paths are similar to those of Case 1.
1 = C (x7 ; x) and P2 = C (x4 ; x6 )yC (xo ; x2 ).
x and P2 = x2p 5 C (x1 ; x2p 6 )yxo x2p 4 .
even, the paths are P1 = y and P2 = C (xk+1 ; xk 1 ).
For k = 2p
w
:
For
For
Fn
4
even, the paths are
y:
Fn
with 0
Case 2 n = 2p and p
xj .
(Fn ). We
onsider two
ases:
C (xk
and ending at
; Fn is hamiltonian virus-free.
Case 1 n = 2p + 1 and p
xi
We follow Remark 1 and Remark 2. In ea
h step of the proof, we show
the paths
all
beginning at
C
y:
P
3 the paths are P1 =
the paths are P1 =
x:
the paths are
Remark 5. For n
P
2 2 6 ) and P2 = xo x1 xx2p 3 x2p 5 x2p 4 .
C (x ; x p
1 = x2p 3
and
P
2 = xo x2p 4 yC (x2 ; x2p 5 )x1 .
8, the hypohamiltonian digraphs
8 is in favor of Conje
ture 5.
Fn
given in Theorem
12
4
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
Con
lusion
It is well known that the problem to de
ide when a digraph is hamiltonian
is NP-
omplete [3℄. A \yes" answer to the hamiltoni
ity problem for a given
digraph
an be veri ed by
he
king in polynomial time that a sequen
e of
verti
es given by an ora
le is a hamiltonian
ir
uit. In
ase of non-hamiltonian
digraphs, as stated in [7℄ pages 28, 29, there is no known way of verifying a
\yes" answer to the
omplementary problem of de
iding if a digraph is nonhamiltonian. A solution to this problem is to provide a hamiltonian virus,
whose presen
e in the digraph
an also be
he
ked in polynomial time. In
ase of the non-hamiltonian hamiltonian virus-free digraphs, they must hold
the parti
ular stru
ture given in Remark 2. The virus notion has been used
in random generation of digraphs without
ertain properties [8℄.
We have built an intera
tive support tool
alled GRAPHVIRUS [5℄ that
allows the graphi
al edition of hamiltonian viruses and the veri
ation that
a given stru
ture is a hamiltonian virus. GRAPHVIRUS
an also be used to
derive a pro
edure for de
iding whether a given digraph is non-hamiltonian.
This pro
edure is of the same
omplexity of the problem of de
iding if a given
digraph is hamiltonian, but the interest of the pro
edure is the fa
t of using
a lo
al stru
ture.
Finally, the theoreti
interest of the results presented here is their relation
with the extension of known suÆ
ient
onditions with the new hamiltonian
virus-free
ondition for the existen
e of hamiltonian
ir
uits.
Referen
es
[1℄ M. R. Brito, W. Fernandez de la Vega, O. Meza, O. Ordaz, Viruses in
Graphs and Digraphs, Vishwa International Journal of Graph Theory, Volume 2, Number 1 (1993), 35{55.
[2℄ Ch. Delorme, O. Ordaz, D. Quiroz, Tools for studying paths and
y
les in
digraphs, Networks 31 (1998), 125{148.
[3℄ Ch. H. Papadimitriou, K. Steiglitz, Combinatorial optimization, Prenti
eHall, New Jersey, 1982.
[4℄ J. L. Fouquet, J. L. Jolivet, Graphes hypohamiltoniens orientes, Colloques
internationaux C.N.R.S. N o 260, Problemes
ombinatoires et theories des
graphes (1976), 149{151.
Hamiltonian virus-free digraphs
13
[5℄ L. Freyss, O. Ordaz, D. Quiroz, J. Yepez, O. Meza, GRAPHVIRUS: Una
herramienta para el tratamiento de fallas en redes, Pro
eedings de la XXIII
Conferen
ia Latinoameri
ana de Informati
a Panel'97, Valparaiso, Chile
(1997), 243{252.
[6℄ L. Freyss, O. Ordaz, D. Quiroz, A method for identifying hamiltonian
viruses, Pro
eedings de la XXIII Conferen
ia Latinoameri
ana de Informati
a Panel'97, Valparaiso, Chile (1997), 231{242.
[7℄ M. R. Garey, D. S. Johnson, Computers and Intra
tability - A guide to
the Theory of NP-Completeness, W. H. Freeman and Company, New York,
1979.
[8℄ F. Losavio, L. E. Marquez, O. Meza, O. Ordaz, La Generation Aleatoire
de Digraphes dans l'Environnement AMDI, Te
hniques et S
ien
e Informatique, Vol. 10, No 6, (1991) 437{446.
[9℄ C. Thomassen, Hypohamiltonian graphs and digraphs, in Theory and Appli
ations of Graphs, Le
ture Notes in Mathemati
s, Vol. 642, eds. Y.
Alavi and D. R. Li
k, Springer, Berlin (1978), 557{571.
iones Matemati
as Vol. 8 No. 1 (2000), pp. 1{13
Hamiltonian Virus-Free Digraphs
Digrafos Libres de Virus Hamiltonianos
Os
ar Ordaz and Leida Gonz
alez (*)
oordazanubis.
iens.u
v.ve)
(
Departamento de Matem
ati
a and
Centro de Ingenier
a de Software Y Sistemas ISYS.
Fa
ultad de Cien
ias. Universidad Central de Venezuela
Ap. 47567, Cara
as 1041-A, Venezuela.
Isabel M
arquez
Universidad Centro
idental Lisandro Alvarado
De
anato de Cien
ias. Departamento de Matem
ati
a.
Barquisimeto, Venezuela.
Domingo Quiroz (
dquirozusb.ve)
Universidad Sim
on Bolivar. Departamento de Matem
ati
a.
Ap. 89000, Cara
as 1080-A, Venezuela.
Abstra
t
A hamiltonian virus is a lo
al
on guration that, if present in a
digraph, forbids this digraph to have a hamiltonian
ir
uit. Unfortunately, there are non-hamiltonian digraphs that are hamiltonian virusfree. Some families of these digraphs will be des
ribed here. Moreover,
problems and
onje
tures related to hamiltonian virus-free digraphs are
given.
Keywords and phrases: digraph, hamiltonian digraph, hamiltonian
virus.
Resumen
Un virus hamiltoniano es una estru
tura lo
al que, estando presente en un digrafo, impide que este tenga un
ir
uito hamiltoniano.
Desafortunadamente, existen digrafos no hamiltonianos sin virus hamiltonianos. Algunas familias de estos digrafos son des
ritas aqu. Mas
aun, se plantean problemas y
onjeturas relativas a digrafos sin virus
hamiltonianos.
Palabras y frases
lave: digrafos, digrafos hamiltonianos, virus hamiltonianos.
Re
eived: 1999/06/04. Revised: 1999/10/28. A
epted: 1999/11/15.
MSC (1991): 05C45, 05C40.
(*) Resear
h partially supported by
CDCH proje
t. No 03-12-3726-98.
GraphVirus
2
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
1
Introdu
tion and terminology
The importan
e of hamiltonian viruses [1, 8℄ is their relation with the \
erti
ation" of non-hamiltonian digraph families, i.e. they are non-hamiltonian
if and only if they have a hamiltonian virus. For example, balan
ed bipartite
digraphs are hamiltonian if and only if they are hamiltonian virus-free. This
paper has multiple goals:
To identify non-hamiltonian digraph families whi
h are hamiltonian
virus-free, and digraph families that are hamiltonian if and only if they
are hamiltonian virus-free.
To
hara
terize digraphs that do not
ontain hamiltonian viruses of a
given order.
To present and dis
uss problems and
onje
tures related to expe
ted
properties of hamiltonian virus-free digraph families.
1.1
Terminology
The terminology des
ribed in what follows is taken textually from [2℄ and it
will be used throughout the whole paper.
Invariants are integer or boolean values that are preserved under isomorphism. We will be using the following invariants, relations between invariants,
theorems and digraph examples. Let D = (V (D); E (D)) be a digraph.
Integer invariants
nodes : number of nodes of a digraph.
ar
s : number of ar
s of a digraph.
alpha2 : maximum size of a set of nodes whi
h indu
es no
ir
uit of length 2.
alpha0 : maximum size of a set of nodes indu
ing no ar
.
woodall : minfd+ (x) + d (y ) : (x; y ) 2
= E (D ); x 6= y g (if alpha2 = 1, then
woodall = 2nodes by
onvention.)
minimum : minfmindegpositive, mindegnegativeg.
mindegpositive minfd+ (x) : x 2 V (D)g.
mindegnegative minfd (x) : x 2 V (D)g.
Boolean invariants
hamiltonian : the digraph
ontains a hamiltonian
ir
uit.
tra
eable : the digraph
ontains a hamiltonian path.
k -
onne
ted : the deletion of fewer than k verti
es always results in a
onne
ted
digraph.
3
Hamiltonian virus-free digraphs
bipartite : its vertex set is partitioned into two subsets
ea
h ar
has one vertex in
X
and another in
Y
and
X
Y
su
h that
.
antisymmetri
: it does not
ontain a
ir
uit of length two.
(1,1)-fa
tor : it
ontains a spanning subdigraph H su
h that
d
1 for all verti
es.
+ (x) = d
H
H (x) =
Relations between invariants
11 :
R31 :
R
2 ^ nodes 4 =) hamiltonian.
antisymmetri
=) ar
s nodes (nodes
1)/2.
minimum
Theorems and
onje
tures
Theorem 51:
D20 .
k -
onne
ted
^ (alpha0
Theorem 64: antisymmetri
=
hamiltonian.
)
)
h+
^
(nodes
6_
D
7_
D
8.
Theorem 78: r-diregular
Digraphs examples
^
6)
^
h
(woodall
r
h)
nodes
a)
)
h+
1)/2
) hamiltonian _
=
D
) hamiltonian _ 5 _
+ 1) =
D
h)
2) =
^ (nodes 2
^ [ar
s nodes(nodes
^ (minimum
^ (nodes= 2
Best result: see
^ ( 2) ^ (minimum
h
Theorem 77: (nodes = 2a + 1)
D
2)
^ ( 5) ^ (minimum
Theorem 67: antisymmetri
2-
onne
ted
=
hamiltonian. Best result: see D20 :
)
) (1,1)-fa
tor.
=
^ (nodes 2
Theorem 65: antisymmetri
hamiltonian.
Theorem 66: antisymmetri
=
hamiltonian.
k)
D
6.
5)
2℄
5_
4
2
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
Hamiltonian viruses
A hamiltonian virus is a lo
al
on guration that, if present in a digraph,
forbids this digraph to have a hamiltonian
ir
uit.
Let H = (V (H ); E (H )) be a proper indu
ed 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 ) : dH (x) = dD (x)g,
is a hamiltonian virus if and only if for every set of disjoint dire
ted paths
q(j)
1
P1 ; : : : ; Pr
overing V (H ) there exists a path Pj = xj : : : x
j , with q(j ) 1
su
h that either x1j 2 T or xjq(j ) 2 T + . The order of a hamiltonian virus
(H; T + ; T ) is de ned as the
ardinality of V (H ).
Theorem 1 ([1℄).
5
Hamiltonian virus-free digraphs
In what follows a 3-uple (H; T + ; T
if there exists in
onvenien
e we identify
D (x)g
+
and
d
=
T
) is present in a digraph
a proper indu
ed subdigraph
D
f 2
H
x
1
V
with
(H ) :
H ),
d
su
h that
H (x)
=
is a hamiltonian virus of a given digraph
6= ;.
T
d
a given digraph, then there exists a digraph
D
+
D (x)g.
D;
In [1℄ we show that if a 3-uple (H; T + ; T
and
T
H
1
D
if and only
isomorphi
to
f 2
=
x
V
(H ) :
+
H+(x) =
Moreover if (H; T
then we must have
T
(for
H
d
+
;T
)
6= ; or
) is not a hamiltonian virus for
D
where (H; T + ; T
) is present
is hamiltonian.
Theorem 2. If a digraph of order n is free of hamiltonian viruses of order h
2
for some h with
h < n; then it has no hamiltonian viruses of order less
than h.
Equivalently, from a hamiltonian virus of order 1
a hamiltonian virus of order
Proof.
h
h
h
+ 1:
Let us reason ab absurdo. Let
1: Assuming
n
1. Let
x
2
V
(D )
n
D
V
D
h
2 we
an build
n
be a digraph of order
n
ontains a hamiltonian virus (H; T + ; T
(H ) and
H1
the subdigraph indu
ed by
+
+
+
D.
It is
lear that the 3-uple (H1 ; T1
r
loss of generality we
an suppose that
P
i
s
x
; T1
2
V
2
) with
i
r)
T
V
(H
[ f g)
x
T
(P1 ): We
onsider three
ases:
1
1
Case 1 P1 = xx1 : : : x1 : In this
ase x1 = T
(2
T1
and
1 = T
+
is present in D . Now, we shall see that (H1 ; T1 ; T1 ) is a hamiltonian virus.
Let P1 ; : : : ; P be a set of disjoint dire
ted paths
overing V (H1 ): Without
in
=
and 2
) of order
. Hen
e
are disjoint dire
ted paths
overing
V
1
P1
=
s
1
i1 =
x1 : : : x1 , P
(H ). Sin
e (H; T
+
;T
)
1
=
q(j) su
h that x1 2 T
is a hamiltonian virus there exists a path Pj = xj : : : xj
j
q
(
j
)
q
(j )
+
+
1
2 T : Therefore xj 2 T1 or xj 2 T1 :
or xj
s = T +. This situation will be treated as
s
1
Case 2 P1 = x1 : : : x1 x. Hen
e x1 2
1
Case 1.
Case 3 P
i+1 : : : xs , P 11
x1
1
i+1
+
=
1
i
x1 : : : x1 xx
i+1 : : : xs .
1
1
1
In this
ase
1
P1
=
1
i
x1 : : : x1 , P2
i (2 i r) are disjoint dire
ted paths
overing V (H ).
Sin
e x 2
= T
[ T and (H; T + ; T ) is a hamiltonian virus, x11 2 T or xs1 2
+
1
= T
there exists a path Pj1 (3 j r + 1) su
h
= T
and x1
T . In
ase x1 2
1 2
q(j 1)
+
that x1
j 1 2 T or xj 1 2 T +. Therefore (H1 ; T1 ; T1 ) is a hamiltonian
virus of order
h.
=
P
A
ontradi
tion.
Corollary 1. If a digraph of order n has hamiltonian viruses then it
ontains
a hamiltonian virus of order n
Proof.
Dire
tly from Theorem 2.
1.
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
6
From Corollary 1 we have:
Remark 1. Let D = (V (D ); E (D )) be a digraph. If for ea
h x
+
+
+
3-uple (H; T
and
f 2
=
T
) with
;T
x
(H ) :
V
d
=
D
hamiltonian virus-free. Moreover, if
x,
=
x; T
D
f 2
Digraph
Corollary 1
x
V
(H ) :
V
H (x)
(D ); the
=
d
+
d
is not a hamiltonian virus for some
x
then for ea
h hamiltonian virus (H; T + ; T
) of
D
we have
x
2
V
(H ):
shows that there exist non-hamiltonian digraphs hamiltonian
D6
virus-free. In
2
D (x)g
H (x) = dD (x)g is not a hamiltonian virus then D is
H
hamiltonian viruses of order 6 are not present. Then by
D6
does not have viruses.
D6
From Theorem 1 we have:
Remark 2. A hamiltonian virus-free digraph D has the following stru
ture:
for ea
h vertex
paths
x
r
the remaining part
In the next lemma
Lemma 1. Let (H; T
+
V
n(
(H )
Then
[
T
(H
Let
Proof.
D [S ℄
+
6 ;
)=
T
S; T
D
x
+
r
)
have that
s)
2
D T+
DT
r r
r
j
1
T
x
j
or x
r:
q(j)
j 2 T + : On the other hand sin
e S \ (T + [ T
Therefore (H
Let us reason ab absurdo.
of order 4 present in
S; T
+
;T
D.
4
;,
2
. Then
.
Let (H; T + ; T
Let us see that 1
)=
) is a hamiltonian virus.
) be a hamiltonian virus
j j 2 and 1 j j 2.
T
+
T
j j 3 (similarly, j j 3). By a simple inspe
tion on the relative
T
+
T
positions of the ar
s in
ar
or (H; T + ; T
H,
we
an dedu
e that either there exists a symmetri
) is not a hamiltonian virus. A
ontradi
tion. Therefore it
j j 1 and j j 1.
T
+
T
disjoint dire
ted paths
overing
(H; T + ; T
ase
S ):
s is a set of disjoint dire
ted paths
overing
Pj (1
D is free of hamiltonian viruses of order
must be
(H
) is a hamiltonian virus, there exists a path
Theorem 3. Let D be an antisymmetri
digraph with minimum
Suppose
V
s
overing V (D[S ℄) we
P +1 ; : : : ; P
P1 ; : : : ; P ; P +1 ; : : : ; P
with
Proof.
0.
is a hamiltonian virus present in D .
(H ). Sin
e (H; T + ; T
2
it must be 1
j
S.
be a set of disjoint dire
ted paths
overing
P1 ; : : : ; P
x.
be a hamiltonian virus present in D . Let S
+
S, d
(x) = d [ ℄ (x) =
[
℄
be su
h that for all x
)
;T
;T
denotes the subdigraph indu
ed by
Then for any set of disjoint dire
ted paths
V
has a
overing by vertex disjoint
su
h that ea
h one of them makes a
ir
uit with
P1 ; : : : ; P
V
If
T
+
=
; then
T
6= ;, hen
e there are
(H ) that do not verify the
onditions for
) to be a hamiltonian virus. A
ontradi
tion. Consider now the
j j = 1 and j j = 2 (the
ase j j = 2 and j j = 1 is similarly
T
+
T
T
+
T
Hamiltonian virus-free digraphs
7
treated). By enumerating several
ases resulting from the relative positions
of the ar
s in H , sets of disjoint dire
ted paths
overing V (H ) do not verify
the
ondition in order that (H; T + ; T ) is a hamiltonian virus. For the
ase
jT +j = jT j = 1 ( or jT +j = jT j = 2), we
an see the existen
e of sets of
disjoint dire
ted paths
overing V (H ) that do not verify the
onditions to be
a hamiltonian virus. By Theorem 2, D is a hamiltonian virus-free digraph of
order 3.
In Theorem 4, we need the following de nition:
De nition 1. [1℄ A 1-
onne
ted virus is a lo
al
on guration that, if present
in a digraph, forbids this digraph to be 1-
onne
ted. Let H = (V (H ); E (H ))
be a proper indu
ed 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) = dD (x)g and T = f+x 2
V (H ) : d (x) = d (x)g, is a 1-
onne
ted virus if and only if V (H ) = T
or
H
D
V (H ) = T .
Theorem 4.
A hamiltonian virus-free digraph is 2-
onne
ted.
Let D = (V (D); E (D)) be hamiltonian virus-free. Let us reason ab
absurdo. Let (H; T + ; T ) be a 1-
onne
ted virus present in D x for some
+
x 2 V (D ) with V (H ) = T : The
ase V (H ) = T
is treated in a similar way.
Let y 2 V (D x) n V (H ): By Remark 2, there exist vertex disjoint paths
P1 ; : : : ; Pr
overing D
y su
h that ea
h one of them makes a
ir
uit with
y . Sin
e V (H ) = T
then for ea
h Pj = x1j : : : xjq(j ) we have x1j 2= V (H ):
Moreover, for ea
h xti 2 V (H ) \ V (Pi ) we have xti 1 2 V (H ). Hen
e x1i 2
V (H ): A
ontradi
tion.
Proof.
3
Hamiltonian virus-free digraph families
In this se
tion we des
ribe non-hamiltonian and hamiltonian virus-free digraph families. There exist non-hamiltonian digraph families with hamiltonian viruses. This fa
t has allowed to derive problems and
onje
tures that
are presented and dis
ussed in this se
tion.
Theorem 5.
Balan
ed bipartite digraphs are hamiltonian if and only if they
are hamiltonian virus-free.
Let D = (X [ Y ; E (D)) be a hamiltonian balan
ed bipartite digraph.
Hen
e D is hamiltonian virus-free (by Theorem 1 ). Let us assume that D
is hamiltonian virus-free. Let x 2 X (similarly dis
ussed for y 2 Y ). By
Proof.
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
8
Remark 2,
D
has a
overing by vertex disjoint paths
x
P1 ; : : : ; Pr
su
h that
) be these
ir
uits.
Sin
e
is a balan
ed bipartite digraph we have j ( )j = 2
(the
ir
uits
have even length), j j =
+
1+ +
1 and j j =
+
+ + .
ea
h one of them makes a
ir
uit with
x.
Let
(1
Ci
D
i
V
X
Therefore
D
n1
n2
r
Ci
nr
ni
Y
n1
n2
nr
is not balan
ed. A
ontradi
tion.
The next remark follows dire
tly from Theorem 77.
Remark 3. There are no non-hamiltonian and hamiltonian virus-free di-
graphs with
= 2 and
minimum
nodes
= 5.
Noti
e that digraph
D5
has
hamiltonian virus.
The only non-hamiltonian and hamiltonian virus-free digraph with
= 3 and
mum
nodes
= 7 is
D6 .
A non-hamiltonian digraph with
= 2minimum + 1
nodes
mini-
9 has hamil-
tonian viruses. By Theorem 77 the only families of digraphs that are nonhamiltonian and where
nodes
= 2minimum + 1
These families have viruses.
9 holds, are
Proposition 1. A hamiltonian virus-free digraph with nodes
D7
5
and
D8 .
is hamilto-
nian.
Proof.
nodes
Let
D
be a hamiltonian virus-free digraph; then
4 then, by
R11 , D
is hamiltonian.
The
ase
minimum
2.
If
= 5 follows
nodes
dire
tly from Remark 3.
Proposition 2. A hamiltonian virus-free digraph with minimum = 2 is tra
eable or hamiltonian.
Proof.
d
Sin
e
minimum
= 2, there exists
x
2
V
(D ) su
h that
+
d
(x) = 2 or
(x) = 2. Then by Remark 2, there exist at most two vertex disjoint paths
overing
x,
D
say
Pi
=
them makes a
ir
uit with
otherwise the path
r (i)
1 2
(1
xi xi : : : xi
x.
2), su
h that ea
h one of
i
If there is only one path then
r (1)
r (2)
1 2
1 2
x1 x1 : : : x1
xx2 x2 : : : x2
makes
D
is hamiltonian,
D tra
eable.
Proposition 3. A hamiltonian virus-free antisymmetri
digraph with nodes
= 6,
7 or 8 is hamiltonian or tra
eable. Moreover, the only hamiltonian non-
hamiltonian virus-free antisymmetri
digraph with
Proof.
Let
woodall
nodes
D
4.
be a hamiltonian virus-free digraph. Then
By Theorem 65, if
= 7 or 8, if
tra
eable.
If
nodes = 7
minimum
minimum
nodes
= 6 then
D
= 2 then, by Proposition 2,
3 then, by Theorem 65,
D
is digraph E X .
minimum
2 and
is hamiltonian.
D
is
For
hamiltonian
is hamiltonian.
or
9
Hamiltonian virus-free digraphs
The following
onje
ture should be true:
Conje
ture 1. A hamiltonian virus-free antisymmetri
digraph is hamiltonian or tra
eable.
Noti
e that digraph EX is non-hamiltonian, but it is tra
eable.
The following
onje
ture should be true:
Conje
ture 2.
with
A hamiltonian virus-free antisymmetri
r 3 and nodes 4r + 1 is hamiltonian.
r diregular digraph
We
an formulate the following remarks for Conje
ture 2: By Theorem
64, the
onje
ture for r = 3 is true when nodes 8. For
ase 9 nodes 13
the hypothesis hamiltonian virus-free perhaps
an be useful. Noti
e that by
Theorem 78, the
onje
ture is true for nodes = 2r + 1. The
onje
ture is true
from Theorem 66 for r = 5 and nodes 15.
D
Problem 1. Let
be an antisymmetri
and hamiltonian virus-free digraph.
Find the greatest positive integer su
h that when ar
s nodes(nodes-1)/2
then
is hamiltonian.
D
x
x
By Theorem 4 and Theorem 67 we have x 2. Moreover the digraph D20
shows that Theorem 67 is the best possible. Noti
e that D20 has hamiltonian
viruses.
D k
x
Problem 2. Let
be a -
onne
ted and hamiltonian virus-free digraph. Find
the greatest integer su
h that when
then
ontains a (1,1)fa
tor.
alpha0 k + x
D
By Theorem 51 we have that x 0. Moreover digraph D20 shows that
this theorem is the best possible. Noti
e that D20 has hamiltonian viruses.
3.1
Hamiltonian virus-free hypohamiltonian digraphs
This se
tion is devoted to study hypohamiltonian hamiltonian virus-free digraphs and those that have hamiltonian viruses. The methods, for building
hypohamiltonian digraphs, established in [9℄ and [4℄ are given. Some
onje
tures related to hamiltonian virus-free and hypohamiltonian digraphs are
dis
ussed.
A digraph D is hypohamiltonian if it has no hamiltonian
ir
uits but every
vertex-deleted subdigraph D v has su
h a
ir
uit.
It is natural to formulate, as in [6℄, the following
onje
tures:
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
10
Conje
ture 3.
Every hamiltonian virus-free non-hamiltonian digraph is hy-
pohamiltonian.
Or the weaker one:
Conje
ture 4.
Every non-hamiltonian vertex-transitive hamiltonian virus-
free digraph is hypohamiltonian.
Conje
ture 5.
Every hypohamiltonian digraph is hamiltonian virus-free.
Noti
e that digraph
D
6
is hypohamiltonian and hamiltonian virus-free.
In [9℄ Thomassen gives a method for obtaining hypohamiltonian digraphs by
forming the
artesian produ
t of
y
les. We give here a short summary of his
results, in order to give some remarks on Conje
tures 3, 4, 5.
Re
all that if
D
1
and
D
2
is the digraph with vertex set
are digraphs then its
(D1 )
V
to (u1 ; u2 ) is present if and only if
1 1
v u
2
E (D
1 ):
v
V
artesian produ
t D
1 D2
(D2 ) su
h that the edge from (v1 ; v2 )
1 = u1 and v2 u2 2 E (D2 ), or v2 = u2 and
The dire
ted
y
le of length
k;
2
k;
is denoted
Ck .
With
this notation Thomassen gives the following theorems:
Theorem 6 ([9℄).
For ea
h k
antisymmetri
digraph.
k
0
3
2 k
3 6k+4
; m
Cmk
; C
Moreover, C
C
1
is a hypohamiltonian
is hypohamiltonian for ea
h
.
Theorem 7 ([9℄).
There is no hypohamiltonian digraph with fewer than six
verti
es, and for ea
h odd m
3
, C
2 Cm
is a hypohamiltonian digraph.
Remark 4. The hypohamiltonian digraphs C3
C
6k+4 with k 0 (Theorem
6) and the hypohamiltonian digraphs given in Theorem 7 verify Conje
ture 5.
However the digraph
C
4 C11 , i.e.,
= 4 and
k
m
= 3 in Theorem 6, refutes
Conje
ture 5. We have proved that the only non-hamiltonian vertex-transitive
digraph whi
h is also hamiltonian virus-free of order 6, is the hypohamiltonian digraph
C
2 C3 :
Whi
h is in favor of Conje
ture 4. Nevertheless the
Conje
ture 4 is false, the digraph
EX
is non-hamiltonian, vertex-transitive,
hamiltonian virus-free and not hypohamiltonian.
In [4℄ Fouquet and Jolivet give the following theorem for obtaining hypohamiltonian digraphs.
Theorem 8 ([4℄).
For ea
h n
s
ribed below is hypohamiltonian.
For n
= 6,
6 = C2 C3 :
F
6;
the digraph Fn
= (V (Fn ); E (Fn ))
de-
11
Hamiltonian virus-free digraphs
= 2p + 1
For n
f
2p
1; k
= 2p
For n
(0
g[ f
odd
=
2 1 ; yg and E (Fn ) =
+4 ; 1 k
xo ; x ; : : : ; x p
2)
p
xk y and yxk ;
and p
f 1
g[f
0 2
Fn )
: V
i
2p:
taken modulo
3 (
2
and p
2 1 xo ; xi xi+1
x p
xk xk
k
p
2
and xk xk
1; k
4
g
even : Ea
h index is
2 1
, Fn is obtained from F p
repla
ing the ar
2 3
2 3
2 4 x2p 3 ;
x
;
x
x;
x
x
;
x
x
;
x
x;
xx
;
x
x
2 3 2p 4 o
2p 3 2p 5 2p 7 2p 3 1
3 o 2p 4 ; x2p 3 x;
xx2p 3 : Ea
h index is taken modulo 2p
2:
In the next theorem let C = xo x1 : : : x2p 1 xo be a
ir
uit. We denote by
xo by the path x p
x p
xxo and adding the following ar
s: x p
x p
C (xi ; xj )
the indu
ed path of
Theorem 9. For ea
h n
Proof.
w
2
8
V
(1
Pi
2) that
over
i
Fn
xk
For
k
For
k
Fn
2
k
the paths are
For
xk
1 = C (x2 ; x3 )x1 x2p 1 xo
4
with 0 2
and
P
2 = C (x4 ; x2p 2 ):
:
k
3.
p
For
k
= 3 the paths are
k
P
P
odd ex
ept 3
Fn
w.
1.
p
k
Fn
and make a
ir
uit with
1 = y and P2 = C (xk+1 ; xk 1 ).
odd, the paths are P1 = C (xk+4 ; xk 4 ) and P2 = C (xk+1 ; xk+3 )y
3 ; xk 1 ).
and
2p
3 the paths are similar to those of Case 1.
1 = C (x7 ; x) and P2 = C (x4 ; x6 )yC (xo ; x2 ).
x and P2 = x2p 5 C (x1 ; x2p 6 )yxo x2p 4 .
even, the paths are P1 = y and P2 = C (xk+1 ; xk 1 ).
For k = 2p
w
:
For
For
Fn
4
even, the paths are
y:
Fn
with 0
Case 2 n = 2p and p
xj .
(Fn ). We
onsider two
ases:
C (xk
and ending at
; Fn is hamiltonian virus-free.
Case 1 n = 2p + 1 and p
xi
We follow Remark 1 and Remark 2. In ea
h step of the proof, we show
the paths
all
beginning at
C
y:
P
3 the paths are P1 =
the paths are P1 =
x:
the paths are
Remark 5. For n
P
2 2 6 ) and P2 = xo x1 xx2p 3 x2p 5 x2p 4 .
C (x ; x p
1 = x2p 3
and
P
2 = xo x2p 4 yC (x2 ; x2p 5 )x1 .
8, the hypohamiltonian digraphs
8 is in favor of Conje
ture 5.
Fn
given in Theorem
12
4
O. Ordaz, L. Gonzalez, I. M
arquez, D. Quiroz
Con
lusion
It is well known that the problem to de
ide when a digraph is hamiltonian
is NP-
omplete [3℄. A \yes" answer to the hamiltoni
ity problem for a given
digraph
an be veri ed by
he
king in polynomial time that a sequen
e of
verti
es given by an ora
le is a hamiltonian
ir
uit. In
ase of non-hamiltonian
digraphs, as stated in [7℄ pages 28, 29, there is no known way of verifying a
\yes" answer to the
omplementary problem of de
iding if a digraph is nonhamiltonian. A solution to this problem is to provide a hamiltonian virus,
whose presen
e in the digraph
an also be
he
ked in polynomial time. In
ase of the non-hamiltonian hamiltonian virus-free digraphs, they must hold
the parti
ular stru
ture given in Remark 2. The virus notion has been used
in random generation of digraphs without
ertain properties [8℄.
We have built an intera
tive support tool
alled GRAPHVIRUS [5℄ that
allows the graphi
al edition of hamiltonian viruses and the veri
ation that
a given stru
ture is a hamiltonian virus. GRAPHVIRUS
an also be used to
derive a pro
edure for de
iding whether a given digraph is non-hamiltonian.
This pro
edure is of the same
omplexity of the problem of de
iding if a given
digraph is hamiltonian, but the interest of the pro
edure is the fa
t of using
a lo
al stru
ture.
Finally, the theoreti
interest of the results presented here is their relation
with the extension of known suÆ
ient
onditions with the new hamiltonian
virus-free
ondition for the existen
e of hamiltonian
ir
uits.
Referen
es
[1℄ M. R. Brito, W. Fernandez de la Vega, O. Meza, O. Ordaz, Viruses in
Graphs and Digraphs, Vishwa International Journal of Graph Theory, Volume 2, Number 1 (1993), 35{55.
[2℄ Ch. Delorme, O. Ordaz, D. Quiroz, Tools for studying paths and
y
les in
digraphs, Networks 31 (1998), 125{148.
[3℄ Ch. H. Papadimitriou, K. Steiglitz, Combinatorial optimization, Prenti
eHall, New Jersey, 1982.
[4℄ J. L. Fouquet, J. L. Jolivet, Graphes hypohamiltoniens orientes, Colloques
internationaux C.N.R.S. N o 260, Problemes
ombinatoires et theories des
graphes (1976), 149{151.
Hamiltonian virus-free digraphs
13
[5℄ L. Freyss, O. Ordaz, D. Quiroz, J. Yepez, O. Meza, GRAPHVIRUS: Una
herramienta para el tratamiento de fallas en redes, Pro
eedings de la XXIII
Conferen
ia Latinoameri
ana de Informati
a Panel'97, Valparaiso, Chile
(1997), 243{252.
[6℄ L. Freyss, O. Ordaz, D. Quiroz, A method for identifying hamiltonian
viruses, Pro
eedings de la XXIII Conferen
ia Latinoameri
ana de Informati
a Panel'97, Valparaiso, Chile (1997), 231{242.
[7℄ M. R. Garey, D. S. Johnson, Computers and Intra
tability - A guide to
the Theory of NP-Completeness, W. H. Freeman and Company, New York,
1979.
[8℄ F. Losavio, L. E. Marquez, O. Meza, O. Ordaz, La Generation Aleatoire
de Digraphes dans l'Environnement AMDI, Te
hniques et S
ien
e Informatique, Vol. 10, No 6, (1991) 437{446.
[9℄ C. Thomassen, Hypohamiltonian graphs and digraphs, in Theory and Appli
ations of Graphs, Le
ture Notes in Mathemati
s, Vol. 642, eds. Y.
Alavi and D. R. Li
k, Springer, Berlin (1978), 557{571.