Abstract
A graceful labeling of a tree with edges is a bijection such that equal to . A spider graph is a tree with at most one vertex of degree greater than . We show that all spider graphs with at most four legs of lengths greater than one admit graceful labeling.
1. Introduction
Labeled graphs form useful models for a wide range of disciplines and applications such as in coding theory, X-ray crystallography, radar, astronomy, circuit design, and communication network addressing [1]. A systematic presentation of diverse applications of graph labeling is presented in [2].
A graceful labeling of a tree is a bijective function from the set of vertices of to the set such that when each edge is assigned the label , the resulting edge labels are distinct. A tree which admits graceful labeling is called a graceful tree. In 1964, Ringel and Rosa (see [3, 4]) gave the famous and unsolved “graceful tree conjecture” which stated that all trees are graceful.
A spider graph is a tree with at most one vertex of degree greater than 2. Gallian [1] has noted that the special case of this conjecture regarding spider graphs is still open and that very few classes of spider graphs are known to be graceful. Huang et al. [5] proved that all spider graphs with three or four legs are graceful. Bahls et al. [6] also proved that every spider graph in which the lengths of any two of its legs differ by at most one is graceful. Jampachon et al. [7] have also proven that is graceful, if is large enough, where is defined in Section 3.
In this paper, we prove that all spider graphs with at most four legs of lengths greater than one are graceful.
2. Preliminaries
Let be a tree with edges. A graceful labeling of is a bijection such that when each edge is assigned the label , the edge label set is equal to . A tree which admits graceful labeling is called a graceful tree.
To prove our results, we need some terminology and existence results which are described below. In [8], Hrnčiar and Haviar proved Lemma 1 and in [9] Jampachon and Poomsa-Ard proved Lemmas 2 and 3.
Lemma 1. Let be a tree with edges and a graceful labeling . Then, the function defined by is also a graceful labeling of .
Lemma 2. Let be a path graph with and let , and . Then, there is a bijective labeling such that or , where and the edge label set is .
Lemma 3. Let , , be a path graph with and let . Then, there is a bijective labeling such that or , where and the edge label set is .
Let be a tree and let be a leaf of . Let be the tree obtained from by adding the vertices and the edges . In [10], Sangsura and Poomsa-Ard have proved Lemma 4.
Lemma 4. If a tree has a graceful labeling such that , where is a leaf of , then is graceful for each .
3. Main Results
A spider graph or spider is a tree with at most one vertex of degree greater than and this vertex is called the branch vertex and is denoted by . A leg of a spider graph is a path from the branch vertex to a leaf of the tree. Let , , denote a spider of legs such that its legs has length one except for legs of lengths , where for all .
Lemma 5. If has a graceful labeling such that, then there is a graceful labeling of such that .
Proof. Let be a graph such as shown in Figure 1(a). Since is a graceful labeling of a subgraph of , extending to such that , , then we get that is a graceful labeling of as shown in Figure 1(b).
(a)
(b)
Theorem 6. The graph is graceful.
Proof. We note that is a path of length ; for example, . Let be a labeling of by labels alternating between the lowest and highest unused numbers in the set . We have that is a graceful labeling of such that and by Lemma 5 we get that is graceful.
Lemma 7. If has a graceful labeling such that, then is also graceful.
Proof. By assumption and Lemma 5, there exists a graceful labeling of such that . From the proof of Lemma 5, we see that is the maximum number; that is, . Let for all vertices of . By Lemma 1, we get that the function is a graceful labeling of such that . Then, is graceful by Lemma 4.
Theorem 8. The graph is graceful.
Proof. From the proof of Theorem 6, we see that the labeling of is a graceful labeling such that . By Lemma 7, we therefore get that is also graceful.
Theorem 9. The graph is graceful.
Proof. Without loss of generality, let and let be the path as shown in Figure 2.
Let be a labeling of by labels alternately between the lowest and highest unused numbers in the set . We have that is a graceful labeling of such that . Note that and . Since if and if , then by Lemma 3 we can find a graceful labeling of such that . Hence, is graceful by Lemma 7.
Theorem 9 is not a new result; it follows from Jampachon and Poomsa-Ard [9], but our proof here is shorter.
Next, consider the path obtained from by deleting the edges and and adding the edges and as shown in Figure 3.
It can be seen that and . If we now introduce a special labeling of which can be used to generate a graceful labeling, it follows that if is even, and if is odd.It can be seen that with the labeling above the labels alternate between the lowest and highest unused numbers in the set . Moreover, we see that admits graceful labeling and that the label of the branch vertex is ; that is, . For convenience, we call the labeling above .
Remark 10. Let be the path obtained from as shown in Figure 3 and let be the labeling of of . Note that , , and . To find a graceful labeling of , we change the values of at the vertices and of such that and .
Next we want to show that are graceful and to prove this result we need the following lemmas.
Lemma 11. If , then there is a graceful labeling of such that .
Proof. Let be the path as shown in Figure 4 and let be the labeling of of . We see that and , where is the leaf of the path . If we change the values of at by reversing their labels, then we get a graceful labeling of such that .
Lemma 12. If and , then there is a graceful labeling of such that .
Proof. Let be the path as shown in Figure 5 and let be the labeling of of . We see that and .
To construct a graceful labeling of , we consider two cases.
Case 1 ( is odd). First, we switch the values of at and , , such that the new value of at is and at is . We then change the new values of at by reversing their labels; then, by Lemma 3, we can find a graceful labeling of such that .
Case 2 ( is even). Define as follows: for , such that for , if is even if is odd, for , if is even if is odd, for , if is even if is odd, for , if is even if is odd for , if is even if is odd, for , if is even if is odd.In accordance with the above labeling pattern, we get a graceful labeling of such that .
Lemma 13. If , then there is a graceful labeling of such that .
Proof. Let be the path as shown in Figure 6 and let be the labeling of of . We see that and .
If is even, then by Lemmas 2 and 3, we can find a graceful labeling of such that . Now suppose that is odd. To construct a graceful labeling of , we must consider two cases.
Case 1 ( is even)
Case 1.1 (). We see that and lies in the path . Change the values of at by reversing their labels; then, the new value of at is . Next, change the new values of at by reversing their labels. Since the number of vertices of the path is even, by Lemmas 2 and 3, we can find a graceful labeling of such that .
Case 1.2 (). Define as follows: For , and for , if is even if is odd, for , if is even if is odd, for , if is even if is odd.In accordance with the above labeling pattern, we get a graceful labeling of such that .
Case 1.3 (). We see that and lies in the path . Change the values of at by reversing their labels. Since the number of vertices of the path is even, by Lemmas 2 and 3, we can find a graceful labeling of such that .
Case 2 ( is odd). Switch the values of at and , ; after that, follow a similar procedure as for Case 1.
Lemma 14. If and , then there is a graceful labeling of such that .
Proof. Let be the path as shown in Figure 3 and let be the labeling of of . We see that and, since , there must be a vertex laying in the path such that . If is even, then by Lemmas 2 and 3, we can find a graceful labeling of such that . Now suppose that is odd.
Case 1 ( is even)
Case 1.1 (). Do similarly as in Case 1.1 of Lemma 13.
Case 1.2 (). Do similarly as in Case 1.3 of Lemma 13.
Case 2 ( is odd). Switch the values of at and , , after which do similarly as in Case 1.
Theorem 15. Let , , , and be integers greater than one. Then, there are three of them; say , , and , for which the spider graph has a graceful labeling such that .
Proof. Without loss of generality, let , let be the path as shown in Figure 5, and let be the labeling of of .
Case 1 (). By Lemma 11 there is a graceful labeling of such that .
Case 2 (). If , then by Lemma 12 there is a graceful labeling of such that . Now suppose that .
If , then by Lemma 13 there is a graceful labeling of such that .
If , then by Lemma 11 there is a graceful labeling of such that .
If and , then by Lemma 12 there is a graceful labeling of such that .
If and , then by Lemma 14 there is a graceful labeling of such that .
If , then by Lemma 13 there is a graceful labeling of such that .
If and , then by Lemma 12 there is a graceful labeling of such that .
If and , then by Lemma 14 there is a graceful labeling of such that .
Case 3 (). By Lemma 13 there is a graceful labeling of such that .
Case 4 (). If , then by Lemma 14 there is a graceful labeling of such that . Now suppose that .
If , then by Lemma 14 there is a graceful labeling of such that .
If , then by Lemma 12 there is a graceful labeling of such that .
If , then by Lemma 14 there is a graceful labeling of such that .
If , then by Lemma 13 there is a graceful labeling of such that .
If , then by Lemma 14 there is a graceful labeling of such that .
If , then by Lemma 13 there is a graceful labeling of such that .
If , then by Lemma 14 there is a graceful labeling of such that .
Theorem 16. The graph is graceful.
Proof. By Theorem 15, there is a graceful labeling of a spider graph with three legs, such that . By Lemma 7, we therefore get that is graceful.
4. Conclusion and Remarks
The main tools required to construct a graceful labeling of are a graceful labeling with , where is the branch vertex, and the results of Lemmas 2 and 3.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This research was financially supported by Udon Thani Rajabhat University, Thailand. The authors would like to thank the Department of Mathematics, Faculty of Science, for other kinds of support and encouragement.