International Journal of Mathematical Sciences and Applications
Vol. 1 No. 2 (May, 2011)
Copyright © Mind Reader Publications
www.ijmsa.yolasite.com
A Class Of Graceful Lobsters with even number of branches incident on
the central path
Debdas Mishra1*,Amaresh Chandra Panda 2†
and Rajani Ballav Dash3
Department of Mathematics
C.V. Raman College of Engineering,Bhubaneswar1,2
Department of Mathematics,Ravenshaw University,Cuttack3
Email:[email protected], [email protected]
Abstract
We observe that a lobster with diameter at least five has a unique path (called the central path) such
that and are adjacent to the centers of at least one where
, and besides adjacencies in the central
path each , is at most adjacent to the centers of some .We call an odd branch if is
odd , an even branch if is non-zero even, and a pendant branch if . In this paper we give graceful labeling to
some new classes of lobsters with each vertex of the central path is attached to an even number of branches.
Furthermore the branches incident on the central path are either even branches or some combinations of even and
pendant branches.
Keywords: graceful labeling, lobsters, component moving transformation, odd and even branches, transfer of the
first type and second type.
AMS classification:05C78
1. Introduction
Definition 1.1 A graceful labeling of a tree with edges is a bijection such that
!" #! " # $%
&'(&
)
A tree which has a graceful labeling is called a
graceful tree.
Definition 1.2 A lobster is a tree having a path from which every vertex has a distance at most two. It is easy to
check that a lobsters * of diameter at least five has a unique path
+ such that besides the
adjacencies in +, each , is at most adjacent to the centers of some stars , whereas
the vertices and are adjacent to the centers of at least one star with
. This path + is called the
central path of the lobsters *.Throughout the paper we use + to denote the central path of a lobster with
diameter at least five. If a vertex , + is adjacent to the center of , then we call an even
branch if is nonzero even, an odd branch if is odd, and a pendant branch if . Therefore, the branches
incident on a vertex in the central path of a lobster can be classified into three types, i.e. even, odd, and pendant
defined as above. Furthermore, whenever we say , for some , is attached to an even number of
branches, we mean a “ non zero” even number of branches unless otherwise stated.
Notation 1.3 In view of Definition 1.2, a combination of branches incident on any can be
represented by a triple - . where - $%'
. represent the number of odd, even, and pendant branches
incident on . We use the symbols & and ) to represent a non-zero even number and an odd number,
respectively. For example :& / means an even number of odd branches, no even branch, and an odd
number of pendant branches.
In 1979, Bermond [1] conjectured that “all lobsters are graceful” which is a special case of the famous and
unsolved “the graceful tree conjecture” of Ringel and Kotzig (1964) [12], which states that “all tress are
464
Amaresh Chandra Panda
graceful”. Bermond’s conjecture is also open and very few classes of lobsters are known to be graceful. Ng
[10], Wang et al. [13], Chen et al. [2] , Morgan [9] (see [3]), and Mishra and Panigrahi[4,5,6,7,8,11] have given
graceful labeling to some classes of lobsters. However, none of the above results gives graceful labeling to a
class of lobsters in which every vertex of the central path is attached to an even number of branches.
The lobsters to which we give graceful labeling in this paper have the following characteristic
feature.
i) The vertex is attached to the combination & &. For an integer 0 , 0 each is attached
to the combination
) ). If 0 1 then each 0 2 03
03 is attached to the combination
& & If 03 , then each
03 2 , is attached to the combination & .
† Please make all correspondence regarding this paper to amareshchandrapanda
ii) The lobsters obtained by deleting one or more combinations in the lobsters of type-I in the above
iii) The vertex is attached to the combination ) ) or & . Each , is attached to the
combination & .
Figure 1: Example of a lobster of type (a). The vertex
is attached to the combination & &. Each of and 3 is attached to ) ), each of 4 and 5 is attached to & & and each of 6 and 7 and 8 is
attached to & 2.
Preliminaries
Definition 2.1 For an edge & " # of a tree , we define " as that connected component of & which
contains the vertex ". Here we say " is a component incident on the vertex
#
. If $ and 9 are vertices of a
tree , " is a component incident on $ and 9
: " then deleting the edge $ " from and making 9
and " adjacent is called the ;)<)%&%0
)#%(
0=$%)=$0)%. Here we say the component " has
been transferred or moved from $ to 9 .This is illustrated in Figure 2. Throughout the paper we write “the
component "” instead of writing “the component "”. Whenever we wish to refer "
as a vertex, we write “
the vertex "”. By the label of the component “"” we mean the label of the vertex "
Figure 2: The tree obtained from the tree by moving the component " from the vertex $ to the vertex
9.
A Class Of Graceful Lobsters…
465
Notation 2.2 For any two vertices $ and 9 of a tree , the notation $ > 9 0=$%&=
means that we move
some components incident on the vertex $ to the vertex . If we consider successive transfers $ > 9 9 >
; ; > ' we simply write $ > 9 > ; > ' transfer. In a transfer $ > $3 > $4 > ? > $@ ,
we call each vertex except $@ a vertex of the transfer.
Lemma 2.3 [8] Let be a graceful labeling of a tree ; Let $ and 9 be two vertices of ; let
"and # be
two components incident on $, where 9 : " A #. Then the following hold:
i)If " 2 # $ 2 9 then the tree B obtained from by moving the components " and #
from $ to 9 is also graceful.
ii) If " $ 2 9 then the tree BB obtained from by moving the component " from $
to 9 is
also graceful.
Definition 2.4 Let be a labelled tree and $ and 9 be two vertices of , and $ be attached to some
components. The $ > 9 transfer is called $
0=$%&=
)
0C&
=0
0-<&
if the labels of the transferred
components constitute a set of consecutive integers. The $ > 9 transfer is called
$
0=$%&=
)
0C&
&;)%'
0-<& if the labels of the transferred components can be divided into two segments,
where each segment is a set of consecutive integers.
Figure 3: The tree in (a) is a tree with a graceful labeling . The trees in (b) and (c) are obtained from (a) by
applying a transfer of the first type D > and a transfer of the second type D > , respectively.
Though the following two results i.e. Lemma 2.5 and Lemma 2.7 already exist in the literature, still we give
their here because we adopt the method described in the proofs of these lemmas to carry out a sequence of the
first and second types for giving graceful labeling to the lobsters here.
Lemma 2.5 [8] In a graceful labeling of a graceful tree , let $ and 9 be the labels of two vertices. Let $ be
attached to a set E of vertices (or components) having labels % % 2 % 2 % 2 <%
,which satisfy
either (i) % 2 2 2 % 2 < $ 2 9 or (ii) % 2 2 % 2 < 2 -
. Then
the following hold.
(a) By making a transfer $ > 9 of the first type we can keep an odd number of components at $ from the
set E and move the rest to 9 and the resultant tree thus formed will be graceful
(b) If E has an even number of elements , then by making a transfer $ > 9 of the second type we can
keep an even number of components at $ from the set E and move the rest to 9, and the resultant tree
thus formed will be graceful.
Proof: (a) By definition of the transfer of the first type, the labels of the components which are transferred to must be consecutive integers. If the elements of E satisfy(i) (respectively, (ii)), then we keep any odd number
of elements , say = 2 elements, namely %
% 2 2 % 2 < = FGHIGJKLMGNO % 2
< % 2 % 2 < = at $ , and transfer the rest to 9.Let E be the set of elements of E
which has been moved to 9. For each . , E , either we have . $ 2 9 or there is another (unique) element
P in E such that . 2 P $ 2 9. Therefore, the new tree thus formed is graceful by Lemma 2.3.
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Amaresh Chandra Panda
(b)Since E contains an even number of elements , < is odd. If the elements of E satisfy (i)
(respectively,(ii)), then from E we keep an even number of elements , say = elements namely %
% 2
QR
QS
% 2 2 % 2 < (respectively, % 2 < and % 2
% 2 % 2 < = if = ;
3
3
(respectively, % 2 < and % 2
) at $, and move the rest to 9 . The
otherwise % and % 2
3
3
transfer that we have done here is easily seen to be a transfer of second type and the resultant tree thus
formed is graceful by Lemma 2.3.
T
Observation 2.6 [8] (a) For any pair of vertex labels and - in a graceful tree, where is attached to a set of
components E as in Lemma 2.5, whenever we make a transfer > - of the first type as in Lemma 2.5(a), the
set E of components of E that are transferred to - is of the form E % 2 = % 2 = 2 % 2 = with
V SVR
% 2 = 2 2 % 2 = 2 -, where U W
X. Further , by re-pairing the elements of E , we get,
QR
QS
3
X,
for U W
3
(1) % 2 = 2 2 % 2 = 2 - and
(2) % 2 = 2 2 2 % 2 = 2 - 2 .
V SVR
Therefore, next if we make a transfer - > or - > 2 , then the set E and the vertices (or labels) and or - and 2 satisfy the hypothesis of Lemma 2.5(a).
(b) We like to mention one important point , i.e. the vertices of the sequences of transfer we deal in this paper
has the property “P”: for any three consecutive integers " # and P of the sequence transfer, we have “P " Y ”. Because of this property “P” we can use Lemma 2.5(a) and part(a) of this observation repeatedly , this
will be clear in Lemma 2.7.
(c) If ,- , and E are as in $ and we make a transfer of second kind > - as in Lemma 2.5(b), then the set
of components of E transferred to - is divided into two disjoint segments, say E and E3 where E and E3 are
of the form E $ $ 2 $ 2 and E3 $3 $3 2 $3 2 with $ 2 2
$3 2 2 - T
Lemma 2.7 [8] In a graceful labeling of a tree , let $ $ $ $ < 9 9 2 9 2 9 2
<3 (respectively,$ $ 2 $ 2 $ 2 = 9 9 9 9 =3 be some vertex labels. Let the
vertex $ be attached to a set E of vertices (or components) having labels % % 2 % 2 % 2 <
(different from the above vertex labels) if and satisfy either % 2 2 2 % 2 < $ 2 9
ZF
% 2 2
QR
% 2 < $ 2 9 U X, then the following hold.
3
(a) By making a sequence of transfers of the first type $ > 9 > $ > 9 2 > $ > 9 2 >
? > (respectively ,$ > 9 > $ 2 > 9 > $ 2 > 9 > ? > where . $ < or 9 2 <3 ( respectively,. $ 2 = or 9 =3 ), an odd number of elements from E can be kept at each
vertex of the transfer and the resultant tree thus formed will be graceful.
(b) If E contains an even number of elements, then by making a sequence of transfers of second type
$ > 9 > $ > 9 2 > $ > 9 2 > ? > . (respectively, $ > 9 > $ 2 > 9 > $ 2 > 9 > ? > ., where . $ < or 9 2 <3 (respectively, . $ 2 = or 9 =3 ), an
even number of elements from E can be kept at each vertex of the transfer, such that the resultant tree
thus formed will be graceful.
(c) Let E contain an odd number of elements. By making a transfer $ > 9 of the first type followed by a
transfer 9 > $ (respectively, 9 > $ 2 of the second type, we can keep from E an odd number
of elements at $ and an even number of elements at 9 and move the rest to $ (respectively,a+1),that
the resultant tree thus formed will be graceful.
Proof: (a) The vertices of the given sequence of transfers satisfy the property P and therefore we can use
Lemma 2.5(a) and observation 2.6(a) repeatedly and hence the proof follows.
(b)By using Lemma 2.5(a) and Observation 2.6(a) repeatedly we can keep an odd number of vertices at
each vertex of transfer $ > 9 > $ > 9 2 > $ > 9 > $ > 9 2 > $ (respectively,
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A Class Of Graceful Lobsters…
$ > 9 > $ 2 > 9 > $ > 9 > $ 2 > 9 > $ 2 ,because the vertices of the sequence
of transfers satisfy the property P .Since each of the vertices $ 9 $ and 9 2 (respectively, $ 9 $ 2 and 9 appear twice in the sequence of transfers and any positive even integer can be written as the sum
of two odd positive integers, the result follows.
(c) The proof follows from the part (b) and Observation 2.6(a)
T
3. Results
In this section we give the main result (i.e. Theorem 3.1) of this paper.
Theorem 3.1 The lobsters in Table 3.1,3.2, and 3.3 given below are Graceful.
Table 3.1
Lobsters [
(a)
& &
Table 3.2
Lobsters [
(a)
(b)
(c)
Table 3.3
Lobsters [
(a)
\ ] %
) )
\ % 2 ] %3
& &
& &
& &
& &
\ ] %
) )
) )
& &
) ) or & \ %3 2 ] & \
% 2 ] & &
& & \ ] & Proof: Let * be a lobsters of type $ in Table 3.1. Figure 1 is an example of such a lobster. We get a graceful
labeling of * if we proceed as per the following steps.
1.
2.
For , let & and < be the number of even and pendant branches, respectively, incident on . Let
!^*! _
a ) _a & ` and _ab < Q .
We first form the graceful tree c* as shown in Figure 4 and Figure 5 with d^ec*f 2 d,i.e. we
attach a new pendant vertex R to the vertex , the degree of each vertex , , is two, and is attached to pendant vertices. We consider the following graceful labeling of c*.
If is even :
# 3 3
3
j
h 2 2 # 3R
3
3 q
# i
= = 2 2 3
3
3
h
gkZF
KlG
IGmnomK
MGFKLJGH
onpoJGmK
KZ
If is odd:
# j
h
S
3
# 3R S
3
S
2 # 3 3
R
3
i = = S 2 S 2 S 3
3
3
h
gkZF
KlG
IGmnomK
MGFKLJGH
onpoJGmK
KZ
q
(3.1)
(3.2)
468
Amaresh Chandra Panda
Figure 4: The tree c* corresponding to the lobsters * for the case is even.
Figure 5: The tree c* corresponding to the lobster * for the case is odd
Let E be the set of all pendant vertices adjacent to in c*. The set E can be written as
$ $3 $rS , where , for ,
E 2 Lk
LH
GMGm
q
$s t 2 Lk
LH
Znn
rSR
X.
Further, the elements of E satisfy $s 2 $rSRSs 2 U
3
& Lk
& LH
GMGm q
3. Let us define an integer u by u v
& Lk
& LH
Znn
w
w
We keep pairs e$s $rSRSs f at and move the rest to and let the tree thus formed
3
3
be c The tree c has the graceful labeling by Lemma 2.3. Let E denote the set of vertices in E that
are transferred to , i.e. E $ x R $ x R3 $rSS x . We define the integers = , as = & if % , and = & z 2 if % 2 , where z are arbitrary non-negative integers
with z 2 1 & . Then we carry out the transfer > 3 > ? > > R with each transfer is
a transfer of first type and keep = vertices from E at each for The resultant tree say, c3
thus formed after the transfer will be graceful by Lemma 2.7.
ER is the set of vertices of E that have come to the vertex R after step 3. We make a transfer
R , i.e. R , of the first type and bring back all the elements of ER to . Then we
remove the vertex R Obviously, the new tree thus formed, say c4 is graceful.
Consider the transfer 3 > S > S3 > ? > @WR > @W with each transfer in 3 is a
transfer of the first type. By Observation 2.6, the labels of the vertices and S and the set ER
satisfy the properties of the vertices $ and 9 and the set E, respectively, of 2.7. So we carry out the transfer
3 consisting of % successive transfer of the first type keeping { 2 vertices from ER at each
vertex of the transfer 3 . By Lemma 2.7 the new tree, say c5 thus formed is graceful. Let |@W be the set
of vertices of ER which have been transferred to @W .
Next we carry out the transfer @W > @WR and transferring every vertex of |@W to @WR . Obviously, the
new tree thus formed, say c6 , is graceful.
y
4.
5.
6.
y
y
469
A Class Of Graceful Lobsters…
7.
8.
9.
Next consider the transfer 4 @WR 2 > @WR 2 > @y > @yR . By Observation 2.6, the
vertices @W R 2 and @WR 2 and the set |@W satisfy the properties of the vertices $ and 9 and the set
E, respectively, of Lemma 2.7. Now we carry out the transfer 4 @WR 2 > @WR 2 > @y >
@yR consisting of %3 % successive transfers of the first type keeping } 2 vertices from |@W at each
vertex of the transfer 4 , where } , for % 2 %3 } 2 1 < are arbitrary integers. By Lemma
2.7, the resultant tree say c7 thus formed after the transfer 4 is graceful. Let ~@y be the set of vertices of
|@W which has been transferred to the vertex @yR after the transfer 4 .
Next carry out transfer @yR > @y bringing all the vertices of ~@y to the vertex @y . Obviously, the new
tree thus formed, say c8 is graceful.
Consider the transfer 5 @y > @yS > . By Observation 2.6, the vertices @y and @yS of 5 and
the set ~@y satisfy the properties of the vertices $ and 9 and the set E, respectively, of Lemma 2.7. For
%3 , we define =B
=B v
< Lk
%
q
< } Lk
% 2 %3
Now we carry out the transfer 5 @y > @yS > , where each transfer is a transfer of the first type
and keep =B vertices at each vertex of the transfer. By Lemma 2.7, the resultant tree say c thus formed
after the transfer 4 is graceful.
10. Let € be the set of vertices of ~@y that have come to the vertex after the transfer 5 . Finally, we consider
the transfer 6 > $ > $rS > $3 > $rSS > ? > $ , where  R
3
Lk
` LH
Znn
omn
2 Lk
` LH
GMGm. By Observation 2.6, we see that the vertices and $ and the set € satisfy
3
the properties of the vertices $ and 9 and the set E, respectively, of Lemma 2.7. Now we carry out the
transfer 6 , where each transfer is a transfer of the second type keeping desired even number of vertices
from € in each vertex of 6 , so that we back the desired lobster. By Lemma 2.7(c), the lobster is graceful.
T

Example: Figure 1 represents a lobster * of type $ in table 3.1. Here ‚
ƒ % %3 „.Here & < & < … &3 … <3 &4 „ <4 &5 <5 „ &6 „ &7 &8 „†
<6 <7 <8 Here ` and Q . We take {4 {5 {6 {7 and {8 † }4 }5 . The method outlined in the steps 2 to 10 in the proof above is described in Figures 6 to
15. Figure 15 represents the lobster * given in figure 1 with a graceful labeling.
Figure 6: The tree c* corresponding to * obtained in step 2. Here ƒ $s … 2 ‡ˆ and
hence E „‚ 470
Amaresh Chandra Panda
Figure 7: The tree c obtained from c* after carrying out the transfer > in step 3.Here & u E $3 $4 $‰8 ‚D Figure 8: The tree c3 obtained from c after carrying out the transfer > 3 > ? > 8 >  in
step 3. Here ER E …„ ‡„ Figure 9: The tree c4 obtained from c3 after carrying out the transfer  > 8 in step 4.
Figure 10: The tree c5 obtained from c4 after carrying out the transfer 3 8 > 7 > 6 > 5 > 4 >
3 in step 5. Here % |@W |3 ‚D ‡
A Class Of Graceful Lobsters…
Figure 11: The tree c6 obtained from c5 after carrying out the transfer 3 > 4 in step 6.
471
Figure 12: The tree c7 obtained from c6 after carrying out the transfer 4 4 > 5 > 6 in step 7. Here
%3 „ ~@y ~3 ƒˆ ˆˆ
Figure 13: The tree c8 obtained from c7 after carrying out the transfer 6 > 5 in step 8.
Figure 14: The tree c obtained from c8 after carrying out the transfer 5 5 > 4 > 3 > > in
step 9. Here € ˆD .
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Amaresh Chandra Panda
Figure 15 : The lobster * in figure 1 with a graceful labeling obtained from the graceful tree c after
carrying out the transfer 6 > $ > $rS > $3 > $rSS > ? > $ ‡… i.e. the transfer
> $ > $‰ > $3 > $‰8 > ? > $7 > $‰4 , in step 10.
For the lobsters in Table 3.2, the proof follows if we make few changes in step 1 to 10 of the proof involving
the lobster of type(a) in Table 3.1. For lobsters of type (a),repeat steps 1,2 and 3, do not remove the vertex
R in step 4, repeat steps 5 and 6, set %3 in steps 7,8and 9, remove the vertex R in step 8, and
repeat step 10. For lobsters of type (b), we repeat steps 1,2,3 and 4, set % and { 2 0 , where
0 < , for % and 0 { 2 for % 2 , in step 5,and set € | ,~@y ER and
replace 5 with 3 in step 10. For lobsters of type (c), repeat steps 1 and 2, set = & { , for , in step 3, repeat step 4,and set % and %3 % in steps 5 to 10. For the lobsters of type (a)in Table
3.3, the proof follows from the proof involving the lobsters of type (a) in Table 3.1 if we repeat steps 1 and
2, set % , i.e.
= & { , for , in step 3, repeat step 4, set
% in step 5, and set
€ | ,~@y ER and replace 5 with 3 in step 10
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