Proof. Since any tree is bipartite, and is the bipartition of vertices of a tree T, where and holding . Clearly, and for each edge of T. By the hypothesis of the theorem, the tree T admits a set-ordered graceful labeling f, without loss of generality, we have with , with and for every edge , obviously, condition is satisfied. We define another set-ordered graceful labeling g of T as follows: with , with , and for every edge . Therefore, we have for every vertex . According to Definition 1, f and g are a matching of graceful v-image-labelings with vertex-image coefficient .
There is another labeling
h of
T defined as:
with
,
with
, since
and
, then
and the vertex label set
, in addition,
for each edge
with
, so the edge label set
, which shows that the labeling
h is just a set-ordered graceful labeling of
T. In addition, we find
for each edge
, immediately, we get
for edge-image coefficient
, see
Figure 2.
The proof of the Theorem 1 is complete. □
Theorem 2. If a tree T admits set-ordered graceful labeling, then T holds the following assertions:
T admits a matching of odd-graceful v-image-labelings and a matching of odd-graceful e-image-labelings.
T admits a matching of odd-elegant v-image-labelings and a matching of odd-elegant e-image-labelings.
T admits a matching of -graceful v-image-labelings, and a matching of -graceful e-image-labelings.
T admits a -arithmetic total labeling and a -arithmetic total labeling are a matching of v-image-labelings; T admits a matching of -arithmetic total e-image-labelings.
T admits a matching of super edge-magic total v-image-labelings and a matching of super edge-magic total e-image-labelings.
Proof. (1) We define a matching of odd-graceful v-image-labelings
and
from the set-ordered graceful v-image-labelings
f and
g. Letting
with
,
with
, and
for each edge
. Since
holds true, then the set of vertex labels of
T under the labeling
is a subset of
, and
For any two vertices
,
since
, we claim that
is an odd-graceful labeling. Then, we define another odd-graceful labeling
by setting
with
,
with
, and
for each edge
. Immediately, the vertex label set
, and the vertex label set is the union of an odd numbers set and an even numbers set, so
we get
for each vertex
and claim that
and
are a matching of odd-graceful v-image-labelings with
.
We also define another odd-graceful labeling by setting with , with , so , for each edge . In addition, , because , so , for distinct . Therefore, is also an odd-graceful labeling, there is for each edge , so and are a matching of set-ordered odd-graceful e-image-labelings with edge-image coefficient . The assertion has been proven.
(2) Setting with , with , since and , so . We set for each edge , we can find is odd, and is even, so is odd, thus , so is an odd-elegant labeling. There is another odd-elegant labeling by setting with , and with , we can see , and . In addition, for each edge , because the parity of and is opposite, so is odd, then we get , we also get for , which indicate that and are a matching of odd-elegant v-image-labelings with .
An odd-elegant labeling of T can be obtained from in the following way: with , with , thus, the vertex label set ; since for each edge , we get the edge label set . Additionally, it is not hard to compute for , thus and are a matching of odd-elegant e-image-labelings with . The assertion has been proven.
(3) For integers
and
, we define a labeling
as:
with
,
with
. Since
and
, we can get
, and the vertex label set under
is a subset of
. Moreover,
which shows that
is a
-graceful labeling of
T. There is another
-graceful labeling
defined as:
with
,
with
, and
for every edge
. We obtain the vertex label set
, and the edge label set
, moreover,
holds true for
, so we claim that
and
are a matching of
-graceful v-image-labelings with
.
Another
-graceful labeling
defined as:
with
,
with
,
for every edge
,
, so
We can get , which shows that and are a matching of -graceful e-image-labelings with edge-image coefficient . We have shown the assertion .
(4) We, by defining a new labeling
, set a transformation:
with
,
with
, then
, so
for distinct
, and
, on the other hand,
for each edge
, we calculate the set
, which means that the labeling
is a
-arithmetic total labeling of
T. Another labeling
differing from
is defined as:
with
,
with
, and
for each edge
, furthermore, the vertex label set
, and
So, is a -arithmetic total labeling, in addition, for each vertex , this means that and are a matching of v-image-labelings with .
We come to define a
-arithmetic total labeling
of
T as follows:
with
,
with
,
for each edge
, the vertex label set
, and moreover
Then, we considering , so labelings and are a matching of -arithmetic total e-image-labelings with edge-image coefficient . The assertion holds true.
(5) There is a labeling
defined in the following way:
with
, and
with
, and we set
for each edge
. It is not difficult to compute
, since
,
and
, we have
, so
is a super edge-magic total labeling with magic coefficient
by definition. We come to define an edge-magic total labeling
in the way:
with
,
with
,
for each edge
, notice that
, so condition
is true, and
to be a constant. Hence,
is a super edge-magic total labeling, moreover,
for each
holds true, immediately, both labelings
and
are a matching of super edge-magic total v-image-labelings with
.
Another set-ordered edge-magic total labeling with magic coefficient is defined as: with , with , for each edge . We also obtain . In addition, there is for each edge , which means that and are a matching of super set-ordered edge-magic total e-image-labelings with edge-image coefficient . This is the proof of the assertion . □
Proof. Let be the bipartition of vertices of a tree T, where and holding . By the hypothesis of the theorem, T admits a graceful labeling f such that with , with and for each edge , as well as and , again, , so f is a set-ordered graceful labeling. Next, we define an edge-magic total labeling l: with , with , and for every edge , as well as , and we have is a positive constant, thus l is called an edge-magic total labeling. In addition, we can see is a constant, so f and l are a matching of e-image-labelings with . □