EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5305 ISSN 1307-5543 – ejpam.com Published by New York Business Global Graceful Labeling of Some Spider Graphs Kittisak Saengsura1,∗, Tiang Poomsa-ard2 1 Algebras and Applications Research Unit, Mahasarakham Universtity, Kantarawichai, Mahasarakham, Thailand 2 Department of Mathematics, Faculty of Science, Khonkaen University, Khonkaen, Thailand Abstract. A graceful labeling of a tree T with n edges is a bijection f : V (T ) −→ {0, 1, 2, . . . n} such that {|f(u) − f(v)| : uv ∈ E(T )} equal to {1, 2, 3, . . . , n}. A spider graph is a tree with one vertex of degree at least 3 and all others with degree at most 2. We show that some classes of spider graphs admit graceful labeling. 2020 Mathematics Subject Classifications: 05C05, 05C15 Key Words and Phrases: Spider graph, graceful labeling, inverse transformation 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, electronic 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 special kind of labeling graph, i.e., a graceful label- ing f of a tree T is a bijective function from the set of vertices V (T ) of T to the set {0, 1, 2, . . . , |E(T )|} such that when each edge xy is assigned the label |f(x) − f(y)|, the resulting edge labels are distinct. There are other concepts of labeling graphs that are equivalent to graceful labeling, for instance, edge antimagic vertex labeling and rainbow antimagic labeling (see [3], [4]). A tree that admits graceful labeling is called a graceful tree. In 1964, Rangel and Rosa (see [5], [6]) gave the famous and unsolved graceful tree conjecture which stated that all trees are graceful. In order to solve the graceful tree conjecture, we also start to consider a graceful labeling of some graph which usually belongs to a subgraph of a tree such as we called a spider graph. The results of graceful labeling on spider graphs is a step towards potentially solving the graceful tree conjecture. A spider graph is a tree with one vertex of degree at least 3 and all others with degree at most 2. Gillian [1] has noted that the special case of the conjecture regarding a spider ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5305 Email addresses: kittisak.s@msu.ac.th (K. Saengsura), tiang.poom@gmail.com (T. Poomsa-ard) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 2 of 8 graph is still open and that very few classes of spider graphs are known to be graceful. Huang et al. [7] proved that all spider graphs with three or four legs are graceful. Bahls et al. [8] 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. [9], [10] have also proven that Sn(k, l,m) is graceful, where Sn(k, l,m) is defined in Section 2. Recently, Panpa et al. [11] proved that all spider graphs with at most four legs of lengths greater than one are graceful. To prove our results, we need some terminology and existence results which are de- scribed below. Let T be a tree with n edges. A graceful labeling of T is a bijection f : V (T ) → {0, 1, 2, . . . , n} such that when each edge xy is assigned the label |f(x) − f(y)|, the edge label set is equal to {1, 2, 3, . . . , n}. A tree with a graceful labeling is called a graceful tree. In [12] Hrnčiar and Haviar proved Lemma 1 and in [10] Jampachon and Poomsa-ard proved Lemma 2 and Lemma 3. Lemma 1. Let T be a tree with n edges and a graceful labeling f . Then, the function f∗ : V (T ) → {0, 1, 2, . . . , n} is given f∗(v) = n− f(v) is also a graceful labeling of T. Lemma 2. Let P2n be a path graph with V (P2n) = {v1, v2, v3, . . . , v2n} and let M = {a+1, a+2, . . . , a+n,m+a+1,m+a+2, . . . ,m+a+n}, m ≥ n and a ≥ 0. Then, there is a bijective labeling f : V (P2n) → M such that f(v1) = i or f(v2n) = i, where i ∈ M and the edge label set is {m− n+ 1,m− n+ 2, . . . ,m+ n− 1}. Lemma 3. Let Pn be a path graph with V (Pn) = {v1, v2, v3, . . . , vn} and let M = {m,m+ 1,m + 2, . . . ,m + n − 1}. Then, there is a bijective labeling f : V (Pn) → M such that f(v1) = i or f(vn) = i, where i ∈ M and the edge label set is {1, 2, 3, . . . , n− 1}. Let T be a tree and let u be a leaf of T . Let T (u, u1, u2, . . . , un) be the tree obtained from T by adding the vertices u1, u2, u3, . . . , un and the edges uu1, u1u2, . . . , un−1un. In [13] Sangsura and Poomsa-ard have proved Lemma 4. Lemma 4. If a tree T has a graceful labeling f such that f(u) = 0, where u is a leaf of T , then T (u, u1, u2, . . . , un) is graceful for n ≥ 1. A spider graph or spider is a tree with at most one vertex of degree greater than 2 and this vertex is called the branch vertex and is denoted by v0. A leg of a spider graph is a path from the branch vertex to a leaf of the tree. Let Sn(m1,m2, . . . ,mk), n ≥ k, denote a spider graph of n legs such that its legs have length one except for k legs of lengths m1,m2, . . . ,mk, where mi ≥ 2 for all i = 1, 2, . . . , k. In [11] Panpa and Poomsa-ard proved Lemma 5 and Lemma 6. Lemma 5. If Sk(m1,m2, . . . ,mk) has a graceful labeling f such that f(v0) = 0, then there is a graceful labeling f ′ of Sn(m1,m2, . . . ,mk) such that f ′(v0) = 0. Lemma 6. If Sk(m1,m2, . . . ,mk) has a graceful labeling f such that f(v0) = 0, then Sn(l,m1,m2, . . . ,mk) is also graceful. K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 3 of 8 In this paper we study graceful labeling of some class of spider graphs whose graceful labeling is generated by our two special labeling types. Now, we consider the special labeling of a graph as the following. Let T be a spider graph of n legs and for each i-th leg of T is represented by a path v − vimi where an integer mi ≥ 2 for all i = 1, 2, . . . , n as shown in Figure 1. It can be seen that |V (T )| = m1 + m2 + m3 + . . . + mn + 1. We now introduce a labeling f of T , which will be used to generate graceful labelings in the following subsequent part. . r rr rr r r ... r r r . . . r rr . . . r r r . . . r r . . . . . . . . . . . . . . .rr rr r rr r rr rr r rr � �� @ @ @ J J J JJ � � � � � � A A A A A A HHH � � � v v11 v51 v31 v21 v12 v52 v32 v22 v1m1 v5m5 v3m3 v2m2 v1(m1−1) v5(m5−1) v3(m3−1) v2(m2−1) v41 v42 v4(m4−1) v4m4 v(n−2)1 v(n−2)2 v(n−2)(mn−2−1) v(n−2)mn−2 v(n−1)1 v(n−1)2 v(n−1)(mn−1−1) v(n−1)mn−1 vn1 vn2 vn(mn−1) vnmn Figure 1 Let T ′ be obtained from T by deleting the edges vv31, vv41, . . . , vv(n−1)1, vvn1 and adding the edges v2m2v31, v3m3v41, . . . , v(n−2)m(n−2) v(n−1)1, v(n−1)m(n−1) vn1. Then we get T ′ is a path. Next we will label the vertices of path T ′ in the following way. Case m1 is odd. Let m1 = 2p + 1. Then we label the vertices of T ′ alternating between the highest and the lowest remaining unused labels by starting with f(v1m1) = m1 +m2 + . . .+mn and f(v1(m1−1)) = o. Then we get f(v) = p. Case m1 is even. Let m1 = 2p′. Then we label the vertices of T ′ alternating between the lowest and the highest remaining unused labels by starting with f(v1m1) = 0 and f(v1(m1−1)) = m1 +m2 + . . .+mn. Then we get f(v) = p′. Notice that the labeling in two cases of T ′ are graceful. We will use the labeling in two cases to generate graceful labeling of some spider graphs in the later part. For convenience, we call the labeling constructed in the first case and the second case as Type I and Type II, respectively. 2. Main Results In this section we want to show that some spider graphs are graceful. At first, for a spider graph that has n legs and there exist n − 2 legs of them are even lengths as the following theorem. Theorem 1. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If mi+1 = 2mi for i = 2, 3, . . . , n− 1, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 4 of 8 Case 1. If m1 is odd, then let f be the labeling of T of Type I. Consider |f(v31) − f(v2m2)|. Since m3 = 2m2, then by the way of labeling of T of Type I there exists v′ ∈ {v31, v32, . . . , v3m3} such that |f(v′) − f(v)| = |f(v31) − f(v2m2)|. Since m3 is even, then by Lemma 2, we can change the labeling f at v31, v32, . . . , v3m3 to be the labeling f ′ such that |f ′(v31) − f(v)| = |f(v31) − f(v2m2)|, the set of vertices labeling of f and the set of vertices labeling of f ′ are the same and the set of edge labeling of f and the set of edge labeling of f ′ are the same. Further, for any i, 4 ≤ i ≤ n consider |f(vi1) − f(v(i−1)mi−1 )|. Since mi = 2mi−1, then by the way of labeling of T of Type I there exists v ′′ ∈ {vi1, vi2, . . . , vimi} such that |f(v′′ )− f(v)| = |f(vi1)− f(v(i− 1)mi−1)|. Since mi is even, then by Lemma 2, we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′ such that |f ′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| the set of vertices labeling of f and the set of vertices labeling of f ′′ are the same and the set of edge labeling of f and the set of edge labeling of f ′′ are the same. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider in a same way as the Case 1, we get T admit graceful labeling. We get the second case as the following theorem. Theorem 2. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If mi+1 = m2 +m3 + . . .+mi + 1, i = 2, 3, . . . , n− 1, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. Case 1. If m1 is odd, then let f be the labeling of T of Type I. Consider |f(v31) − f(v2m2)|. Since m3 = m2 + 1, then by the way of labeling of T of Type I we have |f(v3m3) − f(v)| = |f(v31) − f(v2m2)|. We can change the labeling f at v31, v32, . . . , v3m3 to be the labeling f ′ such that |f ′(v31)− f(v)| = |f(v31)− f(v2m2)| by reversing the order of theirs labels. Further, for any i, 4 ≤ i ≤ n − 1 consider |f(vi1) − f(v(i−1)mi−1 )|. Since mi = m2 + m3 + . . . + mi−1 + 1, then by the way of labeling of T of Type I we have |f(vimi)−f(v)| = |f(vi1)−f(v(i− 1)mi−1)|. We can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′ such that |f ′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| by reversing the order of theirs labels. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider in a similar way as the Case 1, we get T admit graceful labeling. Next we will extend Theorem 1 to be more general. Theorem 3. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If m3,m4, . . . ,mn are even and mi+1 ≥ 2mi for i = 2, 3, . . . , n − 1, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. Case 1. If m1 is odd, then let f be the labeling of T of Type I. Consider |f(v31) − f(v2m2)|. Sincem3 ≥ 2m2, then by the way of labeling of T of Type I we have |f(v3(m2+1))− f(v)| = |f(v31)−f(v2m2)|. Since m3 is even, then by Lemma 2, we can change the labeling f at v31, v32, . . . , v3m3 to be the labeling f ′ such that |f ′(v31)− f(v)| = |f(v31)− f(v2m2)|, the set of vertices labeling of f and the set of vertices labeling of f ′ are the same and K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 5 of 8 the set of edge labeling of f and the set of edge labeling of f ′ are the same. Further, for any i, 4 ≤ i ≤ n consider |f(vi1)− f(v(i−1)mi−1 )|. Let m′ = m2 +m3 + . . .+mi−1. Since mi ≥ 2mi−1 > m′, then by the way of labeling of T of Type I we have |f(vi(m′+1))−f(v)| = |f(vi1)− f(v(i−1)mi−1 )|. Since mi is even, then by Lemma 2, we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′ such that |f ′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| the set of vertices labeling of f and the set of vertices labeling of f ′′ are the same and the set of edge labeling of f and the set of edge labeling of f ′′ are the same. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider similar way as the Case 1 we get T admit graceful labeling. We get the fourth case as the following theorem. Theorem 4. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If m2 = m3 = mn and mi+1 = m2 +m3 + . . .+mi, i = 3, 4, . . . , n− 2, then T is a graceful labeling graph. Proof. Let T be the spider as shown in the Figure 1. Case 1. If m1 is odd, then let f be the labeling of T of Type I. Consider |f(v31) − f(v2m2)|. Since m3 = m2, then by the way of labeling of T of Type I we have |f(v41) − f(v)| = |f(v31)−f(v2m2)|. Further, for any i, 4 ≤ i ≤ n−1 consider |f(vi1)−f(v(i−1)mi−1 )|. Since mi = m2 + m3 + . . . + mi−1, then by the way of labeling of T of Type I we have |f(v(i+1)1) − f(v)| = |f(vi1) − f(v(i− 1)mi−1). Next consider |f(vn1) − f(v(n−1)mn−1 )|. Sincem2 = m3 = mn, then by the way of labeling of T of Type I we have |f(v3m3)−f(v)| = |f(vn1)− f(v(n−1)mn−1 )|. We can change labeling f at v31, v32, . . . , v3m3 to be the labeling f ′ such that |f ′(v31) − f(v)| = |f(vn1) − f(v(n−1)mn−1 )| by reversing the order of theirs labels. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider in a similar way as the Case 1, we get T admit graceful labeling. Next we will extend Theorem 2 to be more general. Theorem 5. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If m3,m4, . . . ,mn are even and mi+1 ≥ m2 + m3 + . . . + mi + 1 for i = 2, 3, . . . , n − 1, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. Case 1. If m1 is odd, then let f be the labeling of T of Type I. Consider |f(v31) − f(v2m2)|. Since m3 ≥ m2 + 1, then by the way of labeling of T of Type I we have |f(v3(m2+1))− f(v)| = |f(v31)− f(v2m2)|. If m3 = m2 + 1, then we can change labeling f at v31, v32, . . . , v3m3 to be the labeling f ′ such that |f ′(v31) − f(v)| = |f(v31) − f(v2m2)| by reversing the order of theirs labels. Suppose that m3 > m2 + 1. Since m3 is even, then by Lemma 2, we can change the labeling f at v31, v32, . . . , v3m3 to be the labeling f ′′ such that |f ′′ (v31) − f(v)| = |f(v31) − f(v2m2)|, the set of vertices labeling of f and the set of vertices labeling of f ′′ are the same and the set of edge labeling of f and K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 6 of 8 the set of edge labeling of f ′′ are the same. Further, for any i, 4 ≤ i ≤ n consider |f(vi1) − f(v(i−1)mi−1 )|. Let m′ = m2 + m3 + . . . + mi−1. Since mi ≥ m′ + 1, then by the way of labeling of T of Type I we have |f(vm′+1) − f(v)| = |f(vi1) − f(v(i−1)mi−1 )|. If mi = m′ + 1, then we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′′ such that |f ′′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| by reversing the order of theirs labels. Suppose thatmi > m′+1. Sincemi is even, then by Lemma 2, we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f iv such that |f iv(vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| the set of vertices labeling of f and the set of vertices labeling of f iv are the same and the set of edge labeling of f and the set of edge labeling of f iv are the same. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider in a similar way as the Case 1, we get T admit graceful labeling. Next we will combine the results of Theorem 2 and Theorem 5 as the following theorem. Theorem 6. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If T satisfies the following conditions: (i) if mi is odd, then mi = m2 +m3 + . . .+mi−1 + 1, (ii) if mi is even, then mi ≥ m2 +m3 + . . .+mi−1 + 1, for any i = 3, 4, . . . , n, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. Case 1. If m1 is odd, then let f be the labeling of T of Type I. For any mi, i = 3, 4, . . . , n consider |f(vi1) − f(v(i−1)mi−1 )|. Let m′ = m2 + m3 + . . . + mi−1. Since mi ≥ m′+1, then by the way of labeling of T of Type I we have |f(vm′+1)−f(v)| = |f(vi1)− f(v(i−1)mi−1 )|. If mi is odd,then by (i) we have mi = m′+1. We can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′ such that |f ′(vi1) − f(v)| = |f(vi1) − f(v(i−1)mi−1 )| by reversing the order of theirs labels. If mi is even, then by (ii) we have mi ≥ m′ + 1. If mi = m′ + 1, then we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′ such that |f ′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| by reversing the order of theirs labels. Suppose thatmi > m′+1. Sincemi is even, then by Lemma 2, we can change labeling f at vi1, vi2, . . . , vimi to be the labeling f ′′′ such that |f ′′′ (vi1)− f(v)| = |f(vi1)− f(v(i−1)mi−1 )| the set of vertices labeling of f and the set of vertices labeling of f ′′′ are the same and the set of edge labeling of f and the set of edge labeling of f ′′′ are the same. Hence T amit graceful labeling. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider in a similar way as the Case 1, we get T admit graceful labeling. Next we will consider a spider graph in which all of its legs except one are equal. Theorem 7. Let T be a spider graph of n legs with lengths m1,m2,m3, . . . ,mn. If m2 = m3 = . . . = mn, then T is a graceful labeling graph. Proof. Let T be the spider as shown in Figure 1. K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 7 of 8 Case 1. If m1 is odd, then let f be the labeling of T of Type I. Since m2 = m3 = . . . = mn, then by the way of labeling of T of Type I for n is odd we have |f(v41)− f(v)| = |f(v31)− f(v2m2)|, |f(v61)− f(v)| = |f(v41)− f(v3m3)|, . . ., |f(v(n−1)1)− f(v)| = |f(v((n−1)/2+1)1)− f(v((n−1)/2)m(n−1)/2 )| and |f(vnmn)− f(v)| = |f(v((n−1)/2+2)1)− f(v((n−1)/2+1)m(n−1)/2+1 )|, |f(v(n−2)mn−2 )−f(v)| = |f(v((n−1)/2+3)1)−f(v((n−1)/2+2)m(n−1)/2+2 )|, . . ., |f(v3m3)−f(v)| = |f(vn1)−f(v(n−1)mn−1 )|. For each i, i = 3, 5, . . . , n, if we change the labeling f at vi1, vi2, . . . , vimi by reversing the order of theirs labels, then we get T is grace- ful. For n is even we have |f(v41)− f(v)| = |f(v31)− f(v2m2)|, |f(v61)− f(v)| = |f(v41)− f(v3m3)|, . . ., |f(vn1)− f(v)| = |f(v(n/2+1)1)− f(v(n/2)mn/2 )| and |f(v(n−1)mn−1 )− f(v)| = |f(v(n/2+2)1)−f(v(n/2+1)mn/2+1 )|, |f(v(n−3)mn−3 )−f(v)| = |f(v(n/2+3)1)−f(v(n/2+2)mn/2+2 )|, . . ., |f(v3m3)−f(v)| = |f(vn1)−f(v(n−1)mn−1 )|. For each i, i = 3, 5, . . . , n−1, if we change the labeling f at vi1, vi2, . . . , vimi by reversing the order of theirs labels, then we get T is graceful. Case 2. If m1 is even, then let f be the labeling of T of Type II and consider similar as the Case 1 we get T admit graceful labeling. 3. Conclusion In this paper, we obtain that certain spider graphs, whose labeling is constructed in the forms of Type I or Type II, admit graceful labeling whenever the lengths of their legs satisfy the conditions presented in Theorems 1 through 7. Acknowledgements This research was financially supported by Mahasarakham University. The authors wish to extend appreciation to the Faculty of Science at Mahasarakham University for providing research facilities. The authors are immensely grateful to the reviewer for their valuable suggestions, which have greatly helped improve this work. References [1] J. A. Gillian. A dynamic survey of graph labeling. The Electronic Journal of Com- binatorics, 16:DS6, 2015. [2] G. S. Bloom and S. W. Golomb. Applications of numbered undirected graphs. Pro- ceedings of the IEEE, 65(4):562–570, 1977. [3] B. J. Septory, L. Susilowati, Dafik, V. Lokesha, and V. Nagamani. On the study of rainbow antimagic connection number of corona product of graphs. European Journal of Pure and Applied Mathematics, 16(1):271–285, 2023. [4] B. J. Septory, L. Susilowati, Dafik, and V. Lokesha. On the study of rainbow antimagic connection number of comb product of friendship graph and tree. Symmetry, 15(1):42, 2023. K. Saengsura, T. Poomsa-ard / Eur. J. Pure Appl. Math, 18 (2) (2025), 5305 8 of 8 [5] G. Ringel. Theory of graphs and its applications. In Proceedings of the Symposium Smolenice, page 162, Prague, 1964. Czechoslovak Academy of Sciences. [6] A. Rosa. On certain valuations of the vertices of a graph. In Theory of Graphs, International Symposium, Rome, July 1966, pages 349–355, New York, 1966. Gordon and Breach. [7] C. Huang, A. Kotzig, and A. Rosa. Further results on tree labellings. Utilitas Math- ematica, 21:31–48, 1982. [8] P. Bahls, S. Lake, and A. Wertheim. Gracefulness of families of spiders. Involve, 3(3):241–247, 2010. [9] P. Jampachon, K. Nakprasit, and T. Poomsa-ard. Graceful labeling of some classes of spider graphs with three legs greater than one. Thai Journal of Mathematics, 12(3):621–630, 2014. [10] P. Jampachon and T. Poomsa-ard. Graceful labeling of spider graphs with three legs of lengths greater than one. Far East Journal of Mathematical Sciences, 100(1):51–64, 2016. [11] A. Panpa and T. Poomsa-ard. On graceful spider graphs with at most four legs of length greater than one. Journal of Applied Mathematics, 2016:5327026, 2016. [12] P. Hrnčiar and A. Haviar. All trees of diameter five are graceful. Discrete Mathemat- ics, 233(1-3):133–150, 2001. [13] K. Saengsura and T. Poomsa-ard. Graceful labeling of the classes of spider graphs with four legs of lengths greater than one Sn(k, l,m, 2). Far East Journal of Mathematical Sciences, 100(2):255–269, 2016.