Indian J. pure appl. Math., 38(5): 445-450, October 2007
c Printed in India.
°
GRACEFUL ARBITRARY SUPERSUBDIVISIONS OF GRAPHS
C HRISTIAN BARRIENTOS1
Department of Mathematics, Clayton State University
Morrow, GA 30260, U.S.A.
E-mail: chr [email protected]
(Received 16 June 2005; after final revision 11 April 2007; accepted 18 April 2007)
In this paper we answer two questions posed by Sethuraman and Selvaraju. Are there any graphs
different from paths whose arbitrary supersubdivisions are graceful? The answer is affirmative;
for the family of y-trees a graceful labeling of any arbitrary supersubdivision is obtained. The
other question, related with arbitrary supersubdivision of stars, asks which graphs that are the
one vertex union of complete bipartite graphs are graceful? Some results were known to the
authors, in particular we extend the answer given by them to any collection of complete bipartite
graphs Kmi ,ni where the greatest common divisor of the different ni ’s is 1.
Key Words: Graceful labeling, graceful graph, chain graph
1. I NTRODUCTION
Let G be a graph of order n and size m, where m + 1 ≥ n. A function f is a graceful labeling
of G if f is an injection from V (G) to the set {0, 1, ..., m} such that when each edge uv of G has
assigned the weight |f (u) − f (v)| , the resulting weights are distinct; in other terms, the induced
weights form the set {1, 2, ..., m}. A graph that admits a graceful labeling is said graceful. The
function g(v) = f (v) + k is a translation of f that induces the same weights.
Let f be a graceful labeling of a graph G; suppose that there exists an integer λ such that for
each edge uv of G, either f (u) ≤ λ < f (v) or f (v) ≤ λ < f (u), then f is said to be an αlabeling. The number λ is said to be the boundary value of f . A graph admitting an α-labeling is
necessarily bipartite and λ is the smaller of the two end-vertices of the edge of weight 1. Among
the class of graphs with α-labelings we can mention caterpillars and complete bipartite graphs.
1
Dedicated to the Memory of F. Harary
446
CHRISTIAN BARRIENTOS
Let G be a graph of size m, if f is an α-labeling of G with boundary value λ, then the function
h : V (G) → {0, 1, ..., m} defined by:
(
λ − f (v),
iff (v) ≤ λ
h(v) =
m + 1 + λ − f (v), iff (v) > λ
is also an α-labeling of G with the same boundary value λ. Another interesting property of αlabelings is that they can be transformed into k-graceful labelings. In fact, let f be an α -labeling
of a graph G, suppose that {A, B} is the bipartition of the vertices of G and that f (v) ≤ λ if v ∈ A
and f (v) > λ if v ∈ B; it is well known (see [4] and [7]) that the function g(v) = f (v) if v ∈ A
and g(v) = f (v) + k if v ∈ B induces the weights k, k + 1, ..., m + k − 1. The function g is called
a k-graceful labeling.
Many of the results about graph labeling are collected and updated regularly in a survey by
Gallian [3]. The reader can consult this survey for more information about the subject. The notation
and terminology used in this paper are taken from [3].
2. A RBITRARY S UPERSUBDIVISONS
In 2001, Sethuraman and Selvaraju [6] introduced a generalization of one of the constructions of
graceful graphs given in 1998 by Burzio and Ferrarese [2]. They called this new method of construction supersubdivisions and proved that arbitrary supersubdivisions of paths are graceful. They
ask the following question: “Are there any graphs different from paths whose arbitrary supersubdivisions are graceful?” They suspected a negative answer for this question. In this section, we show
that the question has an affirmative answer presenting a family of trees whose arbitrary supersubdivisions are graceful.
Let F be a graph with n vertices and m edges. A graph G is said to be a supersubdivision of
F if G is obtained from F by replacing every edge ei of F by a complete bipartite graph K2,ri in
such a way that the end-vertices of ei are merged with the two independent vertices of degree ri of
K2,ri after removing the edge ei from F. In Figure 1, we show two examples of supersubdivisions.
A supersubdivision G of a graph F is said to be arbitrary if every edge of F is replaced by an
arbitrary K2,r (r may vary for each edge arbitrarily).
Let v0 , v1 , ..., vn−1 be the consecutive vertices of the path Pn ; the y-tree Yn is the tree of order
n + 1 whose vertex set is V = {v0 , v1 , ..., vn−1 , vn } and its edge set is E = {ei = vi−1 vi : 1 ≤ i
≤ n − 1} ∪ {en = vn−2 vn }. In other terms, Yn is obtained attaching the vertex vn to the vertex
vn−2 of Pn .
Proposition 1 — Arbitrary supersubdivisions of any y-tree are graceful.
P ROOF: Let G be an arbitrary supersubdivision of Yn . That is, for 1 ≤ i ≤ n, each edge ei of
n
P
Yn is replaced by the complete bipartite graph K2,ri . Thus, the size of G is M = 2
ri and its
order is N = n + 1 +
n
P
i=1
i=1
ri .
GRACEFUL ARBITRARY SUPERSUBDIVISIONS OF GRAPHS
447
F IG . 1. Supersubdivisions of two y-trees.
Without loss of generality, we may assume that in the case where K2,rn−1 is not isomorphic to
K2,rn the following inequality holds rn−1 < rn . Let {Ai , Bi } be the bipartition of the vertex set of
K2,ri , where Ai = {v1Ai , v2Ai } and Bi = {v1Bi , v2Bi , ..., vrBii }
Case 1 : When rn > 1.
Let Mi−1 = i − 1 + 2
n
P
j=i
rj . Now we describe the labeling of K2,ri . When 1 ≤ i ≤ n − 1,
f (vkAi ) = i + k − 2 where 1 ≤ k ≤ 2 and f (vkBi ) = Mi−1 − 2(k − 1) where 1 ≤ k ≤ ri . When
i = n, f (v1Ai ) = n − 2, f (v2Ai ) = n − 2 + rn , and f (vkBi ) = n + 2rn − k − 1.
From the definition of f we have that within K2,ri all the labels are different and the weights
cover the interval [M0 − 2r1 , M0 ] when i = 1, [Mi−1 − 2ri−1 − (i − 2), Mi−1 − (i − 1)] when
2 ≤ i ≤ n − 1, and [1, 2rn ] when i = n. Since the intersection of any pair of these intervals is
empty and the union of all of them is [1, M0 ] we have that f is a graceful labeling of G.
Case 2 : When rn = 1. Notice that in this case rn−1 = 1 too.
Everything is the same except the labeling of K2,r ∼
= K2,1 . Here, f (v An ) = n − 2, f (v An )
1
n
2
= n + 1, and f (v1Bn ) = n; thus the weights 1 and 2 are induced. Therefore, f is a graceful labeling
of G.
In Figure 1 we show an example for each case. In the first case the graphs used are K2,3 , K2,1 ,
K2,5 , K2,1 , and K2,2 . In the second case, the graphs used are K2,2 , K2,3 , K2,1 , K2,4 , K2,1 , and
K2,1 . In both cases, the labeling of K2,rn is a translation of a graceful labeling, while the labeling
n
P
of K2,ri , for 1 ≤ i ≤ n − 1, is a translation of a (1 + 2
rj )-graceful labeling.
j=i+1
In conclusion, paths are not the only graphs for which a graceful arbitrary supersubdivision
exists. We have just proved that y-trees can be arbitrarily supersubdivided. In the next section
448
CHRISTIAN BARRIENTOS
we study supersubdivisions of stars. The terminology has been changed to match the one used by
Sethuraman and Selvaraju in [5].
3. O NE V ERTEX U NION OF C OMPLETE B IPARTITE G RAPHS
In [5], Sethuraman and Selvaraju posed the following question: which graphs that are the one vertex
union of t complete bipartite graphs Kmi ,ni , for any mi and for any ni , 1 ≤ i ≤ t are graceful?
They proved that graphs that are the one vertex union of K2,ni , 1 ≤ i ≤ t are graceful if at least
t − 2 of the K2,ni ’s are non-isomorphic. In this section we extend their result showing that graphs
that are the one vertex union of Kmi ,ni , 1 ≤ i ≤ t are graceful if each of them appears at most twice
in the list and mcd(n1 , n2 , ..., nt ) = 1. An example of the one vertex union of five graphs is shown
in Figure 2.
F IG . 2. Graceful one vertex union graph.
Let S = {Kmi ,ni : 1 ≤ i ≤ t}; for each Kmi ,ni in S, we said that {Ai , Bi } is the bipartition
of its vertices, where Ai = {ui,j : 1 ≤ j ≤ mi } and Bi = {vi,j : 1 ≤ j ≤ ni }. Taking, for each
1 ≤ i ≤ t, one element of Ai and identifying the selected vertices we form a new bipartite graph
t
P
(mk + nk ) − t
G that is the one vertex union of all the elements in S. Observe that G has order
and size
t
P
k=1
k=1
mk nk . Notice that in S there is at most one graph of the form K1,a .
Proposition 2 — Let S = {Kmi ,ni : 1 ≤ i ≤ t}, the graph G formed by the one vertex union of
the graphs in S is graceful if mcd(n1 , n2 , ..., nt ) = 1 and each Kmi ,ni appears at most twice in S.
P ROOF: Suppose first that the elements of S are all non-isomorphic. Let Gi = Kmi ,ni for
1 ≤ i ≤ t. Consider the following labeling of the vertices of Gi : f (ui,j ) = (j−1)ni for 1 ≤ j ≤ mi
t
P
mk nk − (j − 1) for 1 ≤ j ≤ ni . Thus, the labels assigned by f on the vertices of
and f (vi,j ) =
k=i
G are taken from {0, 1, ...,
t
P
k=1
mk nk }. The labels on Ai are the non-negative multiples of ni up to
GRACEFUL ARBITRARY SUPERSUBDIVISIONS OF GRAPHS
449
(mi − 1)ni . Since mcd(n1 , n2 , ..., nt ) = 1, the only label repeated is 0, which is assigned once on
each Ai . Hence, identifying all the vertices labeled 0 we obtain the graph G. Observe that the labels
assigned on the elements of the sets Bi are all distinct.
When f is restricted to Gt it is a graceful labeling and when it is restricted to Gi , for 1 ≤ i ≤
t
P
mk nk )-graceful labeling. Therefore, f is a graceful labeling of
t − 1, it corresponds to a (1 +
k=i+1
G; moreover, it is an α-labeling with boundary value λ = (mt − 1)nt .
Suppose now that S contains two copies of some complete bipartite graphs. In this case, we split
S in two sets S1 and S2 such that within each of them, all the elements correspond to non-isomorphic
graphs. Thus, using on each Si the argument previously presented, we can obtain α-labelings f1
and f2 with boundary values λ1 and λ2 , respectively. In this way, we obtain two bipartite graphs
H1 and H2 , of size h1 and h2 , respectively. Let {AH1 , BH1 } the induced bipartition of the vertices
of H1 . Thus,
(
λ1 − f1 (v),
if v ∈ AH1
g(v) =
h1 + 1 + λ1 − f1 (v), if v ∈ BH1
is also an α-labeling of H1 with boundary value λ1 . Notice that with this new labeling, the common
vertex has assigned the label λ1 . Now, adding the constant h2 to each label assigned on a vertex
of BH1 , the labeling g becomes a h2 -graceful labeling of H1 ; that means that the induced weights
are h2 + 1, h2 + 2, ..., h2 + h1 . Now, to each label assigned for f2 on the vertices of H2 we add
the constant λ1 ; in this way, the common vertex of H2 has label λ1 and the induced weights are
1, 2, ..., h2 . Thus, identifying the vertices with label λ1 we obtain an α-labeling of G with boundary
value λ1 + λ2 . Therefore, G is graceful.
In Figure 2 we present an example of this labeling, where S1 = {K1,3 , K2,5 , K3,4 } and S2 =
{K2,5 , K3,4 }.
4. C ONCLUSION
When a supersubdivision G of a graph F is obtained, each edge of G is replaced by a path of length
2 of K2,m . This fact motivates us to consider a similar problem where the K2,m ’s are substituted by
other graphs.
In the second section we proved that paths are not the only graphs that admit arbitrary supersubdivisions with a graceful labeling, this answers one of the questions of Sethuraman and Selvaraju.
In the same direction we can ask: are paths and y-trees the only graphs that admit arbitrary supersubdivisions with graceful labelings?
In the third section we extended the result given by Sethuraman and Selvaraju about the one
vertex union of t graphs of the form K2,n (for different values of n). In our case, if each complete
bipartite graph is used at most twice in the one vertex union, the resulting graph has an α-labeling.
Moreover, in place of K2,n we used graphs of the form Km,n . The final question in this area is: Do
stars admit arbitrary supersubdivisions with graceful labelings?
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CHRISTIAN BARRIENTOS
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