****Theorem** 3**. Let \(S\) be a spider with legs \(L_1,L_2,...L_s\), and let \(\ell_i=|E(L_i)|\) for each \(i\in [1,s]\). Suppose that \(\ell_{i+1}\geq 2\ell_i+2\) for any \(i\in [2,s-1]\), and \(\ell_{2}\geq 2\ell_1+2\) if \(\ell_2 \not\equiv 1 \pmod 4\) and \(\ell_{2}\geq 2\ell_1+4\) otherwise. Then \(S\) is graceful.
Graceful Labeling of Two Families of Spiders
May 14, 2026
Abstract
A graceful labeling of a graph \(G\) is an injective function \(f : V(G) \to \{0, \ldots, |E(G)|\}\) such that \(\{\,|f(u)-f(v)| : uv \in E(G)\,\} = \{1, \ldots, |E(G)|\}\). If such a labeling exists, then we call \(G\) graceful. Introduced by Rosa in 1967, graceful labeling has been widely studied, and the Graceful Tree Conjecture asserts that every tree is graceful. The conjecture is known to hold for several classes of trees, including caterpillars, trees with at most four leaves, trees of diameter at most five, and certain spiders. An important subclass is that of \(\alpha\)-labelings, where a graceful labeling \(f\) admits an integer \(\alpha\) such that each edge joins a vertex with label at most \(\alpha\) to one with label greater than \(\alpha\). A result from 1982 by Huang, Kotzig, and Rosa shows that if \(H\) has an \(\alpha\)-labeling with a vertex \(u\) labeled \(0\) or \(\alpha\), and \(G\) has a graceful labeling with a vertex \(v\) labeled \(0\), then identifying \(u\) and \(v\) yields a graceful graph, though this requires a \(0\)-labeled vertex in \(G\). We prove a related result that relaxes this condition: if \(G\) has a graceful labeling \(f\) such that \(f(u)+\lfloor n/2 \rfloor + 1 \le n\) and \(n \not\equiv 1 \pmod{4}\), where \(u\in V(G)\) and \(n\ge 2\) is an integer, then joining \(u\) to an end vertex of the vertex-disjoint \(n\)-vertex path \(P_n\) yields a graceful graph. As an application, we show that any spider with legs \(L_1,\ldots,L_s\) (\(s \ge 1\)) satisfying \(|E(L_{2})| \ge 2|E(L_1)|+ 4\) and \(|E(L_{i+1})| \ge 2|E(L_i)|+ 2\) for \(i \in \{2,\ldots, s-1\}\) is graceful. Furthermore, we give an explicit graceful labeling for spiders with one leg of arbitrary length and all others of length at most two such that the center is labeled by \(0\). This labeling enables the construction of larger graceful spiders by attaching paths at the center.
Keywords: Graceful labeling; \(\alpha\)-labeling; Graceful Tree Conjecture.
1 Introduction↩︎
For two integers \(p\) and \(q\), we let \([p,q]=\{i\in \mathbb{Z}: p \le i\le q\}\). A graceful labeling of a graph \(G\) is an injective function \(f : V(G) \to [0,|E(G)|]\) such that \(\{\,|f(u)-f(v)| : uv \in E(G)\,\} = [1,|E(G)|]\). If \(G\) admits a graceful labeling, then \(G\) is graceful.
Graceful labelings were first introduced by Rosa [1] under the name of \(\beta\)-valuations, and the term “graceful labeling” was due to Golomb [2]. The major open problem on graceful labelings is the Graceful Tree Conjecture, proposed by Rosa in 1967 [1]. It is also known as the Kotzig–Ringel–Rosa Conjecture, reflecting its close connection to the Ringel Conjecture [3] and Kotzig’s earlier work on related graph decomposition and labeling problems, including special cases of the conjecture.
****Conjecture** 1** (Graceful Tree Conjecture). Every tree is graceful.
While Conjecture 1 has attracted considerable attention, it has been verified only for certain special classes of trees, including caterpillars [1] (a caterpillar is a tree with the property that the removal of all leaves leaves a path), trees with at most four end-vertices [4], and trees with diameter at most five [5]. For additional classes of trees known to admit graceful labelings, we refer the reader to Gallian’s dynamic survey [6]. In this paper, our focus is on graceful spiders.
A spider is a tree with at most one vertex (called the center) of degree greater than two. Let \(S\) be a spider such that the degree of the center is at least three (otherwise, the spider is simply a path). Each path in \(S\) connecting the center and a leaf is called a leg of \(S\). The general question whether all spiders are graceful is still open, but some classes of spiders have been shown to be graceful. Bahls, Lake, and Wertheim [7] proved that spiders whose leg lengths differ by at most one are graceful. Panpa and Poomsa-ard [8] showed that all spiders with at most four legs of length greater than one admit a graceful labeling, and this result was extended by Panpa, Imnang, and Wasuanankul [9] to spiders with at most five such legs. For additional classes of spiders known to admit graceful labelings, we refer the reader to Gallian’s dynamic survey [6].
A special type of graceful labeling, called an \(\alpha\)-labeling, has been used as a constructive tool; see, for example, Panpa and Poomsa-ard [8]. A graceful labeling \(f\) is said to be an \(\alpha\)-labeling if there exists an integer \(\alpha\) such that \(f(u) \leq \alpha < f(v)\) for each edge \(uv\) with \(f(u) < f(v)\). The integer \(\alpha\) is called the index of \(f\). Clearly, a graph must be bipartite in order to admit an \(\alpha\)-labeling. The following lemma of Huang, Kotzig, and Rosa [4] provides a useful method for constructing larger graceful graphs from smaller ones.
****Lemma** 1** ([4]). Let \(G\) and \(H\) be vertex-disjoint graphs with \(u \in V(G)\) and \(v \in V(H)\). If \(G\) admits an \(\alpha\)-labeling with index \(\alpha\) such that \(u\) is labeled by \(\alpha\) or \(0\), and \(H\) admits a graceful labeling such that \(v\) is labeled by \(0\), then the graph obtained from \(G\) and \(H\) by identifying \(u\) and \(v\) is graceful.
The graceful labeling in Lemma 1 is obtained by shifting the vertex labels of \(H\) upward by \(\alpha\), and, when \(u\) is labeled \(\alpha\), shifting the labels in the upper part of the \(\alpha\)-labeling of \(G\) upward by \(|E(H)|\). If instead \(u\) is labeled \(0\), one first applies the reverse \(\alpha\)-labeling of \(G\), thereby obtaining another \(\alpha\)-labeling in which \(u\) receives the label \(\alpha\). The same construction then applies. In either case, the vertex formed by identifying \(u\) and \(v\) is assigned the label \(\alpha\).
For \(n \geq 1\), let \(P_n\) denote a path on \(n\) vertices. Cattell [10] proved that every path \(P_n\) admits an \(\alpha\)-labeling in which a prescribed vertex can be assigned any label in \([0,n-1]\), provided that \(n \not\equiv 1 \pmod{4}\). Consequently, if a spider admits a graceful labeling in which its center is labeled by \(0\), then one can construct a larger graceful spider by attaching a suitable path at its center via Lemma 1. However, after this operation, the center of the resulting spider is no longer labeled by \(0\); it is labeled by \(\alpha\).
Our first contribution addresses this issue by allowing iterative attachment of paths to the center of a graceful spider, provided the path length is relatively large. Let \(G\) and \(H\) be vertex-disjoint graphs with \(u \in V(G)\) and \(v \in V(H)\). We define \(G(u)\oplus H(v)\) to be the graph obtained from the disjoint union of \(G\) and \(H\) by adding the edge \(uv\). We now state our result.
****Theorem** 2**. Let \(G\) be a graph with \(u \in V(G)\), and let \(P_n\) be a path vertex-disjoint from \(G\) with \(v\) an endvertex of \(P_n\). Suppose that \(G\) admits a graceful labeling \(f\) such that \[f(u) + \left\lfloor \frac{n}{2} \right\rfloor + 1 \leq n,\] and that \(n \not\equiv 1 \pmod{4}\). Then \(G(u)\oplus P_n(v)\) admits a graceful labeling \(h\) such that \[h(w) = f(w) + \left\lfloor \frac{n}{2} \right\rfloor \quad \text{for all } w \in V(G).\]
As an application of Theorem 2, we obtain the following result.
Saengsura and Poomsa-Ard [11] proved a result similar as Theorem 3 where they required \(\ell_{i+1}\geq 2\ell_i+2\) but \(\ell_i\) is even for all \(i\in [3,s]\).
Furthermore, we give an explicit graceful labeling for spiders with one leg of arbitrary length and all others of length at most two such that the center is labeled by \(0\).
****Theorem** 4**. Let \(S\) be a spider with one leg of arbitrary length and all others of length at most two. Then \(S\) has a graceful labeling such that its center is labeled by zero.
As an application of Lemma 1 and Theorem 4, we have the following result.
****Theorem** 5**. If \(S\) is a spider with all but three legs of length at most \(2\), then \(S\) is graceful.
The remainder of this paper is organized as follows. In the next section, we prove Theorems 2 and 3. The proofs of Theorems 4 and 5 are given in the last section.
2 Proof of Theorems 2 and 3↩︎
Let \(G\) be a graph and \(f:V(G) \to [0,|E(G)|]\). For \(uv\in E(G)\), define \(f(uv) =|f(u)-f(v)|\). For \(X\subseteq V(G)\) and \(F\subseteq E(G)\), we let \(f(X)=\{f(x): x\in X\}\) and \(f(F) =\{f(e): e\in F\}\). For an integer \(\alpha\) and a set \([p,q]\) with \(p,q\in \mathbb{Z}\), we let \(\alpha+[p,q]=\{\alpha+i: i\in [p,q]\}\). We will need the following result on graceful labelings of paths.
****Lemma** 2**. Let \(P_n\) be a path and \(v\in V(P_n)\).
(a) Then \(P_n\) has a graceful labeling such that \(v\) is labelled by \(0\). (See Cattell [10].)
(b) Then \(P_n\) has an \(\alpha\)-labeling \(f\) such that \(f(v) = 0\) except \(n=5\) and \(v\) is the central vertex. (See Rosa [12] and Cattell [10].)
(c) If \(v\) is an endvertex, then \(P_n\) has an \(\alpha\)-labeling \(f\) such that \(f(v)=i\) for any \(i\in [0,n-1]\) unless \(n=4s+1\) for some \(s\in \mathbb{N}\) and \(i \in \{s, 3s\}\). (See Cattell [10].)
Proof of Theorem 2. Since \(n\not\equiv 1 \pmod 4\) and \(f(u)+\lfloor\frac{n}{2}\rfloor \le n-1 =|E(P_n)|\), by Lemma 2(c), \(P_n\) has an \(\alpha\)-labeling \(g\) with the index \[\begin{align} \alpha &=\lfloor\frac{n}{2} \rfloor -1, \quad \text{and} \\ g(v) &=f(u)+\lfloor\frac{n}{2}\rfloor. \end{align}\]
Let \(m=|E(G)|\) and \(G'=G(u)\oplus P_n(v)\). Note that \(E(G')=E(G) \cup E(P_n) \cup \{uv\}\). We now define a labeling \(h\) of \(G'\) as follows: \[\begin{align} h(w) &= f(w)+\lfloor \frac{n}{2} \rfloor, &&\text{for all } w\in V(G);\\ h(x) &= g(x), &&\text{for all } x \in V(P_{n})\text{ such that } 0\le g(x)\le \lfloor\frac{n}{2}\rfloor-1;\\ h(x) &= g(x)+m+1, &&\text{for all } x \in V(P_{n})\text{ such that } \lfloor\frac{n}{2}\rfloor\le g(x)\le n-1. \end{align}\] Figure 1 illustrates the construction of \(h\).
We first verify that \(h\) is injective from \(V(G')\) to \([0,m+n]\). Since \(f: V(G) \to [0,m]\) and \(g: V(P_n) \to [0, n-1]\) are both injective, by the definition of \(h\), we have \[\begin{align} h(V(G)) &=\lfloor \frac{n}{2} \rfloor+[0,m]=\left[\lfloor \frac{n}{2} \rfloor,m+\lfloor \frac{n}{2}\rfloor\right], \\ h(V(P_n)) &= [0,\lfloor\frac{n}{2}\rfloor-1] \cup \left(m+1+[\lfloor\frac{n}{2}\rfloor, n-1]\right) \\ &= [0,\lfloor\frac{n}{2}\rfloor-1] \cup \left[m+\lfloor\frac{n}{2}\rfloor+1, m+n \right]. \end{align}\] Thus \(h\) is an injective function from \(V(G')\) to \([0,m+n]\).
Next we verify that \(h(E(G'))=[1,m+n]\). As \(h(x) = f(x)+\lfloor \frac{n}{2} \rfloor\) for all \(x\in V(G)\) and \(f(E(G))=[1,m]\), we have \(h(E(G))=[1,m]\). As \(g\) is an \(\alpha\)-labeling of \(P_n\), by the definition of \(h\), we have \(h(E(P_n)) =m+1+[1,n-1] =[m+2, m+n]\). Furthermore, we have \(h(u) =f(u)+\lfloor \frac{n}{2} \rfloor\) and \(h(v) =g(v)+m+1=f(u)+\lfloor\frac{n}{2}\rfloor+m+1\) since \(g(v)\ge \lfloor\frac{n}{2}\rfloor\), and so \(h(uv) =m+1\). Therefore, \[h(E(G')) =[1,m] \cup \{m+1\} \cup [m+2, m+n] =[1,m+n].\]
The above shows that \(h\) is a graceful labeling of \(G'\). The second part of the statement is clear by the definition of \(h\). ◻
Proof of Theorem 3. Let \(S\) be a spider with legs \(L_1,L_2,\ldots, L_s\), and let \(\ell_i=|E(L_i)|\) such that \(\ell_{i+1}\geq 2\ell_i+2\) for any \(i\in [2,s-1]\), and \(\ell_{2}\geq 2\ell_1+2\) if \(\ell_2 \not\equiv 1 \pmod 4\) and \(\ell_{2}\geq 2\ell_1+4\) otherwise. We may assume that \(s\ge 3\), as we are done by Lemma 2(a) otherwise. If there exists \(i\in [2,s]\) such that \(\ell_i \equiv 1 \pmod 4\), then we let \(k_1, \ldots, k_r\) be all the indices with \(k_1<k_2< \ldots <k_r\) such that \(\ell_{k_j} \equiv 1 \pmod 4\) for all \(j\in [1,r]\).
We let \(x\) be the center of \(S\), and let \(xy_i \in E(L_i)\) be the edge of \(L_i\) incident with \(x\). For each \(j\in [1,r]\), we further let \(y_{k_j}z_{k_j}\) be the edge of \(L_{k_j}-x\) that is incident with \(y_{k_j}\). We let \[S_1=S[V(L_1) \cup \{y_{k_j}: j\in [1,r]\}].\] By Lemma 2(a), \(S[V(L_1 )]\) has a graceful labeling \(f_1'\) such that \(x\) is labeled by \(0\). We now define a labeling \(f_1\) for \(S_1\):
f_1(u)= f_1’(u) & ,
_1+j & .
Since \(f_1'(x)=0\) and \(f_1'\) is a graceful labeling of \(S[V(L_1 )]\), it is clear that \(f_1\) is a graceful labeling of \(S_1\).
Let \[L_i^*=L_i-V(S_1) \text{ \quad for each i\in [2,s]. }\] Our goal now is to extend \(f_1\) to a graceful labeling of \(S\) by iteratively adding \(L_i^*\) back to \(S_1\) applying Theorem 2. Let
S_i= S_i-1(x)L^*_i(y_i) & ,
S_i-1(y_i)L^*_i(z_i) & .
Let \(i\in [2,s]\) and suppose that we have found a graceful labeling \(f_{i-1}\) of \(S_{i-1}\) such that \[\label{eqn:label-upper-bound-S1} f_{i-1}(w) \le f_1(w)+\sum^{i-1}_{j=2} \frac{\ell_j}{2} \quad \text{for any w\in V(S_1)}.\tag{1}\] Note that 1 holds for \(f_1\), and that for any \(j\in [2,i-1]\), we have \[\begin{align} \label{eqn:upper-bound-li} \ell_j \le \frac{\ell_{j+1}-2}{2} \le \ldots \le \frac{1}{2^{i-j}}\ell_i-2\sum_{h=1}^{i-j} \frac{1}{2^h} =\frac{1}{2^{i-j}}\ell_i-2(1-\frac{1}{2^{i-j}}). \end{align}\tag{2}\] We consider two cases regarding whether \(i\in \{k_1, \ldots, k_r\}\).
Case 1: \(i\not\in \{k_1, \ldots, k_r\}\).
In order to apply Theorem 2, it suffices to show that \(f_{i-1}(x)+ \lfloor\frac{\ell_i}{2} \rfloor +1 \le \ell_i\). As \(f_1(x)=0\), by 1 , we have \[\begin{align} & f_{i-1}(x)+ \lfloor\frac{\ell_i}{2} \rfloor +1 \\ & \le \sum^{i-1}_{j=2} \frac{\ell_j}{2}+ \lfloor\frac{\ell_i}{2} \rfloor +1 \\ & \overset{\eqref{eqn:upper-bound-li}}{\le} \sum^{i-1}_{j=2} \frac{1}{2} \left( \frac{1}{2^{i-j}}\ell_i-2(1-\frac{1}{2^{i-j}})\right)+\lfloor\frac{\ell_i}{2} \rfloor +1 \\ &\le \ell_i \sum_{j=1}^{i-1} \frac{1}{2^j} - \sum^{i-1}_{j=2}(1-\frac{1}{2^{i-j}}) +1\\ &= (1-\frac{1}{2^{i-1}}) \ell_i -(i-2)+ (1- \frac{1}{2^{i-2}}) +1. \end{align}\] When \(i\ge 4\), it is clear that \(f_{i-1}(x)+ \lfloor\frac{\ell_i}{2} \rfloor +1 \le \ell_i\). When \(i\in [2,3]\), since \(\ell_i \ge 2\ell_{1}+2 \ge 4\), we have \(\frac{1}{2^{i-1}} \ell_i \ge 1\). Thus \(f_{i-1}(x)+ \lfloor\frac{\ell_i}{2} \rfloor +1 \le \ell_i\).
We let \(f_i\) be the graceful labeling of \(S_i\) given by Theorem 2. By the second part of the theorem, we have \(f_i(w) =f_{i-1}(w)+ \lfloor \frac{\ell_i}{2}\rfloor\), and so \(f_i\) satisfies 1 .
Case 2: \(i\in \{k_1, \ldots, k_r\}\).
Suppose that \(i=k_p\) for some \(p\in [1,r]\). In this case, we have \(f_1(y_i) =\ell_1+p\) by the definition of \(f_1\). Note that \(\ell_1\le \frac{1}{2}(\ell_2-4)\) and for each \(j\in [2,i-1]\), we still have 2 .
To apply Theorem 2, we show below that \(f_{i-1}(y_i)+ \lfloor\frac{\ell_i-1}{2} \rfloor +1 \le \ell_i-1\).
As \(f_1(y_i) =\ell_1+p\), by 1 , we have \[\begin{align} & f_{i-1}(y_i)+ \lfloor\frac{\ell_i-1}{2} \rfloor +1 \\ & \le \ell_1+p+\sum^{i-1}_{j=2} \frac{\ell_j}{2}+ \lfloor\frac{\ell_i-1}{2} \rfloor +1 \\ & \overset{\eqref{eqn:upper-bound-li}}{\le} \frac{1}{2}(\ell_2-4)+p+ \sum^{i-1}_{j=2} \frac{1}{2} \left( \frac{1}{2^{i-j}}\ell_i-2(1-\frac{1}{2^{i-j}})\right)+\lfloor\frac{\ell_i-1}{2} \rfloor + 1\\ & \overset{\eqref{eqn:upper-bound-li}}{\le} \frac{1}{2^{i-1}} \ell_i-(1-\frac{1}{2^{i-2}})-2+p+\ell_i \sum_{j=1}^{i-1} \frac{1}{2^j} - \sum^{i-1}_{j=2}(1-\frac{1}{2^{i-j}}) +\frac{1}{2}\\ &= \frac{1}{2^{i-1}} \ell_i-(1-\frac{1}{2^{i-2}})-2+p+(1-\frac{1}{2^{i-1}}) \ell_i -(i-2)+ (1- \frac{1}{2^{i-2}}) +\frac{1}{2}\\ & = \ell_i -2+p-(i-2)+ \frac{1}{2} \\ &= \ell_i-(i-p)+\frac{1}{2}. \end{align}\] Since \(i=k_p\) and \(p\le i-1\), we get \(f_{i-1}(y_i)+ \lfloor\frac{\ell_i-1}{2} \rfloor +1 \le \ell_i- \frac{1}{2}\), which implies \(f_{i-1}(y_i)+ \lfloor\frac{\ell_i-1}{2} \rfloor +1 \le \ell_i- 1\) as \(f_{i-1}(y_i)+ \lfloor\frac{\ell_i}{2} \rfloor +1\) is an integer.
We let \(f_i\) be the graceful labeling of \(S_i\) given by Theorem 2. By the second part of the theorem, we again have \(f_i(w) =f_{i-1}(w)+ \lfloor \frac{\ell_i-1}{2}\rfloor\), and so \(f_i\) satisfies 1 .
Continue the process above, we can find a graceful labeling \(f_s\) for the spider \(S\). ◻
3 Proof of Theorems 4 and 5↩︎
In this section, we prove Theorems 4 and 5. Since the proof of Theorem 5 is shorter when Theorem 4 is assumed, we begin with the latter.
Proof of Theorem 5. Let \(S\) be a spider with all but three legs of length at most \(2\). If \(S\) has at most one leg of length greater than two, then we are done by Theorem 4. Thus, we assume that \(S\) has at least two legs of length greater than two. Let the center of \(S\) be \(x\) and \(L_1\) and \(L_2\) be two legs of \(S\) that each has length at least three. Let \(S^*=S-(V(L_1\cup L_2) \setminus \{x\})\). By Theorem 4, \(S^*\) has a graceful labeling \(g\) such that \(g(x)=0\). By Lemma 2(b), \(L_1\cup L_2\) has an \(\alpha\)-labeling \(h\) with \(h(x)=0\). Now, by Lemma 1, \(S\) has a graceful labeling. ◻
Proof of Theorem 4. It suffices to construct the desired graceful labeling for spiders in which all but at most one leg has length \(2\). Indeed, suppose a spider \(S'\) has a graceful labeling \(f'\) with center \(x\) labeled \(0\). Let \(S^*\) be the spider obtained from \(S'\) by adding \(t\) new vertices \(y_1,\ldots,y_t\) and \(t\) new edges \(xy_1,\ldots,xy_t\). Extend \(f'\) to a labeling \(f^*\) by setting \(f^*(u)=f'(u)\) for every \(u\in V(S')\) and \[f^*(y_i)=|E(S')|+i \qquad \text{for } i\in [1,t].\] Then \(f^*\) is clearly graceful, and \(f^*(x)=0\). Therefore, for the remainder of the proof, let \(S\) be a spider with center \(x_0\) such that all but at most one leg have length \(2\). We construct a graceful labeling \(f\) of \(S\) satisfying \(f(x_0)=0\). Since a spider with exactly two legs is a path, by Lemma 2(a), we suppose that \(S\) has at least three legs.
We let \(N\) be the leg of \(S\) that has length larger than 2 if it exists or let it be an arbitrary leg of \(S\), and let \(L_1, \ldots, L_s\) be all the other legs of \(S\), which are all of length 2, where \(s\ge 2\) is an integer. Let \[\begin{align} L_i&=x_0 u_iv_i \quad i\in [1,s], \\ N&= x_0x_1\ldots x_\ell \quad \text{where \ell =|E(N)|}, \quad \text{and}\\ m&= 2s+\ell. \end{align}\] Note that \(m=|E(S)|\). See Figure 2 for an illustration of the vertex names of \(S\) when \(s=2\) and \(\ell=8\).
We define a labeling \(f: V(S) \to [0,m]\) in two cases according to the parity of \(\ell\).
Suppose \(\ell\) is odd.
For any \(i\in [1,s]\), let \[\begin{align} f(u_{i})&=m-(2i-1),\\ f(v_{i})&=2i-1; \end{align}\] and for any \(i\in [0,\ell]\), let \[\begin{align} f(x_i)&=\begin{cases} 2\times \frac{i}{4} =\frac{i}{2} & \text{if i\equiv 0 \pmod 4},\\ m- 2\times \frac{i-1}{4}=m-\frac{i-1}{2}& \text{if i\equiv 1 \pmod 4},\\ 2s-1 + 2\times \frac{i+2}{4} =2s-1 + \frac{i+2}{2}& \text{if i\equiv 2 \pmod 4}, \\ m-(2s-1) -2\times \frac{i+1}{4} =m-(2s-1) -\frac{i+1}{2}& \text{if i\equiv 3 \pmod 4}. \end{cases} \end{align}\] See Figure 3 for an illustration of \(f\) when \(s=2\) and \(\ell=11\).
Suppose \(\ell\) is even.
For any \(i\in [1,s]\), let \[\begin{align} f(u_{i})&=m-2(i-1),\\ f(v_{i})&=2i-1; \end{align}\] and for any \(i\in [0,\ell]\), let \[\begin{align} f(x_i)&=\begin{cases} 2\times \frac{i}{4}=\frac{i}{2} & \text{if i\equiv 0 \pmod 4}, \\ m-2s -2\times \frac{i-1}{4} = m-2s -\frac{i-1}{2} & \text{if i\equiv 1 \pmod 4},\\ 2s-1 + 2\times \frac{i+2}{4} = 2s-1 + \frac{i+2}{2}& \text{if i\equiv 2 \pmod 4}, \\ m-1-2\times \frac{i-3}{4} = m-1-\frac{i-3}{2} & \text{if i\equiv 3 \pmod 4}. \end{cases} \end{align}\] See Figure 4 for an illustration of \(f\) when \(s=2\) and \(\ell=10\).
In the remainder of the proof, we verify that \(f\) is a graceful labeling of \(S\). Before proceed to the proof, we define a partition of \(V(S)\) and \(E(S)\) as follows. Let \[\begin{align} U &= \{u_i: i\in [1,s]\}, \\ V &= \{v_i: i\in [1,s]\}, \quad \text{and}\\ W_j&=\{x_i: i\in [1,\ell],\, i \equiv j \pmod 4\}, \quad j\in [0,3]; \end{align}\] and \[\begin{align} E_1 &= \{x_0 u_i: i\in [1,s]\}, \\ E_2 &= \{u_iv_i: i\in [1,s]\}, \\ F_j &=\{x_ix_{i+1}: i\equiv j \pmod 4\}, \quad j\in [0,3]. \end{align}\]
We separate the proof into two cases. In both cases, we show that \(f\) is injective from \(V(S)\) to \([0,m]\) and \(f(E(S))= [1,m]\)
Case 1: \(\ell\) is odd.
****Claim** 1**. The function \(f\) is injective from \(V(S)\) to \([0,m]\).
Proof. Since \(m\) is odd in this case, by the definition of \(f\), we know that for any \(v\in V(S)\),
f(v) 0 & ,
1 & .
Thus \(f(u) \ne f(v)\) if \(u\in U \cup W_0\cup W_3\) and \(v\in V \cup W_1\cup W_2\). Therefore, to show that \(f(u) \ne f(v)\) for distinct \(u, v \in V(S)\), we consider \(u,v \in U \cup W_0\cup W_3\) or \(u,v\in V \cup W_1\cup W_2\).
Consider first that \(u,v \in U \cup W_0\cup W_3\). Since \(f(u_i) >f(u_j)\) when \(i<j\) and \(u_i,u_j\in U\), \(f(x_i)<f(x_j)<f(u_s)\) when \(i<j\) and \(x_i,x_j\in W_0\), and \(f(u_s)>f(x_i)>f(x_j)\) when \(i<j\) and \(x_i,x_j\in W_3\), it is left to show that \(f(u) \ne f(v)\) when \(u\in W_0\) and \(v\in W_3\). Let \(u=x_i\) and \(v=x_j\) for \(i\equiv 0 \pmod 4\) and \(j\equiv 3 \pmod 4\) such that \(i,j\in [0,\ell]\). If \(f(u)=f(v)\), then we get \(\frac{i}{2}=m-(2s-1) -\frac{j+1}{2}\). Since \(m=2s+\ell\), this implies that \(i+j=2\ell +1\), a contradiction to \(i,j\in [0,\ell]\).
Consider then that \(u,v \in V \cup W_1\cup W_2\). Since \(f(v_i) <f(v_j)\) when \(i<j\) and \(v_i,v_j\in V\), \(f(x_i)>f(x_j)>f(v_s)\) when \(i<j\) and \(x_i,x_j\in W_1\), and \(f(v_s)<f(x_i)<f(x_j)\) when \(i<j\) and \(x_i,x_j\in W_2\), it is left to show that \(f(u) \ne f(v)\) when \(u\in W_1\) and \(v\in W_2\). Let \(u=x_i\) and \(v=x_j\) for \(i\equiv 1 \pmod 4\) and \(j\equiv 2 \pmod 4\) such that \(i,j\in [1,\ell]\). If \(f(u)=f(v)\), then we get \(m-\frac{i-1}{2}=(2s-1) +\frac{j+2}{2}\). This implies that \(m=2s-1+\frac{i+j-1}{2}\), or equivalently \(i+j=2(\ell+1)+1\), a contradiction to \(i,j\in [1,\ell]\). 0◻
****Claim** 2**. We have \(f(E(S))= [1,m]\).
Proof. Since \(f\) is injective and so \(f(e)\in [1,m]\) for any \(e\in E(S)\), it suffices to show that \(f(e_1) \ne f(e_2)\) for distinct \(e_1, e_2 \in E(S)\).
Since \(m\) is odd, by the definition of \(f\), we know that for any \(e\in E(S)\),
f(e) 0 & ,
1 & .
Thus \(f(e_1) \ne f(e_2)\) if \(e_1\in E_1 \cup F_1\cup F_3\) and \(e_2\in E_2 \cup F_0\cup F_2\). Therefore, to show that \(f(e_1) \ne f(e_2)\) for distinct \(e_1, e_2 \in E(S)\), we consider two collections of six subcases each:
Figure 5: 
.
Figure 6: 
.
We list the expression of \(f(e)\) for \(e\in E(S)\): \[\begin{align} f(e)&=\begin{cases} m-(2i-1)-0=m-2i+1 & \text{if e=x_0u_i \in E_1}, \\ |m-(2i-1)-(2i-1)| = |m-4i+2| & \text{if e=u_iv_i\in E_2},\\ m-\frac{i+1-1}{2}-\frac{i}{2}=m-i& \text{if e=x_ix_{i+1} \in F_0}, \\ m-\frac{i-1}{2}-(2s-1+\frac{i+1+2}{2})=m-2s-i& \text{if e=x_ix_{i+1}\in F_1}, \\ |m-(2s-1)-\frac{i+1+1}{2}-(2s-1+\frac{i+2}{2})|=|m-4s-i|& \text{if e=x_ix_{i+1}\in F_2}, \\ m-(2s-1)-\frac{i+1}{2}-\frac{i+1}{2}=m-2s-i& \text{if e=x_ix_{i+1} \in F_3}. \\ \end{cases} \end{align}\]
Case 1e.1: \(e_1, e_2\in E_1\).
Let \(e_1=x_0u_i\) and \(e_2=x_0u_j\) for some distinct \(i,j\in [1,s]\). Then \[f(e_1) =m-2i+1 \ne m-2j+1 =f(e_2).\]
Case 1e.2: \(e_1, e_2\in F_1\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 1 \pmod 4\) and \(i,j\in [1,\ell-1]\). Then \[f(e_1) = m-2s-i\ne m-2s-j =f(e_2).\]
Case 1e.3: \(e_1, e_2\in F_3\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 3 \pmod 4\) and \(i,j\in [1,\ell-1]\). Then \[f(e_1) = m-2s-i \ne m-2s-j=f(e_2).\]
Case 1e.4: \(e_1\in E_1\) and \(e_2\in F_1\).
Let \(e_1=x_0u_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 1 \pmod 4\) with \(j\in [1,\ell-1]\). Since \(m=2s+\ell\), we have \[f(e_1) =m-2i+1 \ge m-2s+1 \ne m-2s- j =f(e_2).\]
Case 1e.5: \(e_1\in E_1\) and \(e_2\in F_3\).
Let \(e_1=x_0u_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 3 \pmod 4\) with \(j\in [1,\ell-1]\). Since \(m=2s+\ell\), we have \[f(e_1) =m-2i+1 \ge m-2s+1 \ne m-2s- j =f(e_2).\]
Case 1e.6: \(e_1\in F_1\) and \(e_2\in F_3\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i \equiv 1 \pmod 4\), \(j \equiv 3 \pmod 4\) and \(i,j\in [1,\ell-1]\). Then \[f(e_1) =m-2s-i \ne m-2s-j =f(e_2).\]
Case 1o.1: \(e_1, e_2\in E_2\).
Let \(e_1=u_iv_i\) and \(e_2=u_jv_j\) for some \(i,j\in [1,s]\). Suppose, without loss of generality, that \(i<j\). Then \(f(e_1)=|m-4i+2|\) and \(f(e_2) = |m-4j+2|\). Suppose to the contrary that \(f(e_1)=f(e_2)\). Since \(i\ne j\) and \(i<j\), we must have \(4j-2-m= m-4i+2\). This implies that \(2m =2(2i+2j-2)\) or \(m=2i+2j-2\), a contradiction to the assumption that \(m\) is odd.
Case 1o.2: \(e_1, e_2\in F_0\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 0\pmod 4\) and \(i,j\in [0,\ell-1]\). Then \[f(e_1) = m-i \ne m-j =f(e_2).\]
Case 1o.3: \(e_1, e_2\in F_2\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i,j \equiv 2\pmod 4\) and \(i,j\in [1,\ell-1]\). Suppose, without loss of generality, that \(i<j\). Then we have \[\begin{align} f(e_1) =|m-4s-i| \quad \text{and}\quad & f(e_2) =|m-4s-j|. \end{align}\] Since \(i<j\), \(f(e_1) =f(e_2)\) implies that \(4s+j-m=m-4s-i\), and so \(m=4s+\frac{i+j}{2}\). Since \(i,j \equiv 2\pmod 4\), \(\frac{i+j}{2}\) is even and so \(m\) is even. However, this gives a contradiction to the assumption that \(m\) is odd.
Case 1o.4: \(e_1\in E_2\) and \(e_2\in F_0\).
Let \(e_1=u_iv_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 0 \pmod 4\) with \(j\in [0,\ell-1]\). Then \(f(e_1)=|m-4i+2|\) and \(f(e_2) =m-j\). If \(f(e_1) =f(e_2)\), then we either have \(j=4i-2\) or \(m=2i-1+\frac{j}{2}\). In the former case, we get a contradiction to the fact that \(j \equiv 0 \pmod 4\); and in the latter case, we get a contradiction to \(m=2s+\ell\).
Case 1o.5: \(e_1 \in E_2\) and \(e_2\in F_2\).
Let \(e_1=u_iv_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 2 \pmod 4\) with \(j\in [1,\ell-1]\). Then \(f(e_1)=|m-4i+2|\) and \(f(e_2) =|m-4s-j|\). If \(f(e_1) =f(e_2)\), then we have either \(m=2s+2i+\frac{j-2}{2}\) or \(4i=4s+j+2\). In the former case, since \(j \equiv 2 \pmod 4\) and so \(\frac{j-2}{2}\) is even, we get a contradiction to the fact that \(m\) is odd; and in the latter case, we get a contradiction to the fact that \(i\in [1,s]\).
Case 1o.6: \(e_1 \in F_0\) and \(e_2\in F_2\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i \equiv 0\pmod 4\), \(j \equiv 2\pmod 4\), and \(i,j\in [0,\ell-1]\). Then \(f(e_1) = m-i\) and \(f(e_2) =|m-4s-j|\). If \(f(e_1) =f(e_2)\), then we have either \(m= 2s+\frac{i}{2} +\frac{j}{2}\) or \(i=4s+j\). In the former case, since \(m=2s+\ell\), we get \(i+j=2\ell\), a contradiction to \(i,j\in [1,\ell-1]\). In the latter case, we get a contradiction to the fact that \(i \equiv 0\pmod 4\) and \(j \equiv 2\pmod 4\). 0◻
Combining Claims 1 and 2, we know that \(f\) is a graceful labeling of \(S\) when \(\ell\) is odd.
Case 2: \(\ell\) is even.
****Claim** 3**. The function \(f\) is injective from \(V(S)\) to \([0,m]\).
Proof. Since \(m\) is even in this case, by the definition of \(f\), we know that for any \(v\in V(S)\),
f(v) 0 & ,
1 & .
Thus \(f(u) \ne f(v)\) if \(u\in U \cup W_0\cup W_1\) and \(v\in V \cup W_2\cup W_3\). Therefore, to show that \(f(u) \ne f(v)\) for distinct \(u, v \in V(S)\), we consider \(u,v \in U \cup W_0\cup W_1\) or \(u,v\in V \cup W_2\cup W_3\).
Consider first that \(u,v \in U \cup W_0\cup W_1\). Since \(f(u_i) >f(u_j)\) when \(i<j\) and \(u_i,u_j\in U\), \(f(x_i)<f(x_j)<f(u_s)\) when \(i<j\) and \(x_i,x_j\in W_0\), and \(f(u_s)>f(x_i)>f(x_j)\) when \(i<j\) and \(x_i,x_j\in W_1\), it is left to show that \(f(u) \ne f(v)\) when \(u\in W_0\) and \(v\in W_1\). Let \(u=x_i\) and \(v=x_j\) for \(i\equiv 0 \pmod 4\) and \(j\equiv 1 \pmod 4\) such that \(i,j\in [0,\ell]\). If \(f(u)=f(v)\), then we get \(\frac{i}{2}=m-2s-\frac{j-1}{2}\). Since \(m=2s+\ell\), this implies that \(i+j=2\ell +1\), a contradiction to \(i,j\in [0,\ell]\).
Consider then that \(u,v \in V \cup W_2\cup W_3\). Since \(f(v_i) <f(v_j)\) when \(i<j\) and \(v_i,v_j\in V\), \(f(x_j)>f(x_i)>f(v_s)\) when \(i<j\) and \(x_i,x_j\in W_2\), and \(f(v_s)<f(x_j)<f(x_i)\) when \(i<j\) and \(x_i,x_j\in W_3\), it is left to show that \(f(u) \ne f(v)\) when \(u\in W_2\) and \(v\in W_3\). Let \(u=x_i\) and \(v=x_j\) for \(i\equiv 2 \pmod 4\) and \(j\equiv 3 \pmod 4\) such that \(i,j\in [1,\ell]\). If \(f(u)=f(v)\), then we get \(2s-1+\frac{i+2}{2}=m-1-\frac{j-3}{2}\). This implies that \(i+j=2\ell+1\), a contradiction to \(i,j\in [1,\ell]\). 0◻
****Claim** 4**. We have \(f(E(S))= [1,m]\).
Proof. Since \(f\) is injective and so \(f(e)\in [1,m]\) for any \(e\in E(S)\), it suffices to show that \(f(e_1) \ne f(e_2)\) for distinct \(e_1, e_2 \in E(S)\).
Since \(m\) is even, by the definition of \(f\), we know that for any \(e\in E(S)\),
f(e) 0 & ,
1 & .
Thus \(f(e_1) \ne f(e_2)\) if \(e_1\in F_1 \cup E_0\cup E_2\) and \(e_2\in F_2 \cup E_1\cup E_3\). Therefore, to show that \(f(e_1) \ne f(e_2)\) for distinct \(e_1, e_2 \in E(S)\), we consider two collections of six subcases each:
Figure 7: 
.
Figure 8: 
.
We list the expression of \(f(e)\) for \(e\in E(S)\): \[\begin{align} f(e)&=\begin{cases} m-2(i-1)-0=m-2i+2 & \text{if e=x_0u_i \in E_1}, \\ |m-2i+2-(2i-1)| =|m-4i+3| & \text{if e=u_iv_i \in E_2},\\ m-2s-\frac{i+1-1}{2}-\frac{i}{2}=m-2s-i& \text{if e=x_ix_{i+1} \in F_0}, \\ |m-2s-\frac{i-1}{2}-(2s-1+\frac{i+1+2}{2})|=|m-4s-i|& \text{if e=x_ix_{i+1} \in F_1}, \\ m-1-\frac{i+1-3}{2}-(2s-1+\frac{i+2}{2})=m-2s-i& \text{if e=x_ix_{i+1} \in F_2}, \\ m-1-\frac{i-3}{2}-\frac{i+1}{2}=m-i& \text{if e=x_ix_{i+1} \in F_3}. \\ \end{cases} \end{align}\] Case 2e.1: \(e_1, e_2\in E_1\).
Let \(e_1=x_0u_i\) and \(e_2=x_0u_j\) for some distinct \(i,j\in [1,s]\). Then \[f(e_1) =m-2i+2 \ne m-2j+2 =f(e_2).\]
Case 2e.2: \(e_1, e_2\in F_0\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 0 \pmod 4\) and \(i,j\in [0,\ell]\). Then \[\begin{align} f(e_1) =m-2s-i\ne m-2s-j=f(e_2) \end{align}\]
Case 2e.3: \(e_1, e_2\in F_2\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 2 \pmod 4\) and \(i,j\in [2,\ell]\). Then \[f(e_1)= m-2s-i \ne m-2s-j=f(e_2).\]
Case 2e.4: \(e_1\in E_1\) and \(e_2\in F_0\).
Let \(e_1=x_0u_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 0 \pmod 4\) with \(j\in [0,\ell]\). Since \(m=2s+\ell\), we have \[f(e_1) =m-2i+2 \ge m-2s+2 \ne m-2s- j=f(e_2).\]
Case 2e.5: \(e_1\in E_1\) and \(e_2\in F_2\).
Let \(e_1=x_0u_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 2\pmod 4\) with \(j\in [2,\ell]\). Since \(m=2s+\ell\), we have \[f(e_1) =m-2i+2 \ge m-2s+2 \ne m-2s- j =f(e_2).\]
Case 2e.6: \(e_1\in F_0\) and \(e_2\in F_2\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i \equiv 0 \pmod 4\), \(j \equiv 2 \pmod 4\) and \(i,j\in [0,\ell]\). Then \[f(e_1) =m-2s-i \ne m-2s-j =f(e_2).\]
Case 2o.1: \(e_1, e_2\in E_2\).
Let \(e_1=u_iv_i\) and \(e_2=u_jv_j\) for some \(i,j\in [1,s]\). Suppose, without loss of generality, that \(i<j\). Then \(f(e_1)=|m-4i+3|\) and \(f(e_2) = |m-4j+3|\). Suppose to the contrary that \(f(e_1)=f(e_2)\). Since \(i\ne j\) and \(i<j\), we must have \(4j-3-m= m-4i+3\). This implies that \(2m =4i+4j-6\) or \(m=2i+2j-3\), a contradiction to the assumption that \(m\) is even.
Case 2o.2: \(e_1, e_2\in F_1\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i,j \equiv 1 \pmod 4\) and \(i,j\in [1,\ell]\). Suppose, without loss of generality, that \(i<j\). Since \(m=2s+\ell\), we have \[\begin{align} f(e_1) =|m-4s-i| \quad \text{and}\quad & f(e_2) =|m-4s-j|. \end{align}\] Since \(i<j\), \(f(e_1) =f(e_2)\) implies that \(4s+j-m=m-4s-i\), and so \(m=4s+\frac{i+j}{2}\). Since \(i,j \equiv 1\pmod 4\), \(\frac{i+j}{2}\) is odd and so \(m\) is odd. However, this gives a contradiction to the assumption that \(m\) is even.
Case 2o.3: \(e_1, e_2\in F_3\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some distinct \(i,j \equiv 3\pmod 4\) and \(i,j\in [3,\ell]\). Then \[\begin{align} f(e_1) =m-i\ne m-j=f(e_2) \end{align}\]
Case 2o.4: \(e_1\in E_2\) and \(e_2\in F_1\).
Let \(e_1=u_iv_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 1 \pmod 4\) with \(j\in [1,\ell]\). Then \(f(e_1)=|m-4i+3|\) and \(f(e_2) =|m-4s-j|\). If \(f(e_1) =f(e_2)\), then we either have \(j=4(i-s)-3\) or \(m=2s+2i+\frac{j-3}{2}\). In the former case, since \(i\le s\), we get a contradiction to \(j\in [1,\ell]\). In the latter case, since \(j \equiv 1 \pmod 4\) and so \(2s+2i+\frac{j-3}{2}\) is odd, we get a contradiction to the fact that \(m\) is even.
Case 2o.5: \(e_1 \in E_2\) and \(e_2\in F_3\).
Let \(e_1=u_iv_i\) and \(e_2=x_jx_{j+1}\) for some \(i\in [1,s]\) and \(j \equiv 3 \pmod 4\) with \(j\in [3,\ell]\). Then \(f(e_1)=|m-4i+3|\) and \(f(e_2) =m-j\). If \(f(e_1) =f(e_2)\), then we have either \(j=4i-3\) or \(m=2i+\frac{j-3}{2}\). In the former case, we get a contradiction to the fact that \(j\equiv3\pmod 4\); and in the latter case, since \(2i+\frac{j-3}{2} <2s+\ell\), we get a contradiction to the fact \(m=2s+\ell\).
Case 2o.6: \(e_1 \in F_1\) and \(e_2\in F_3\).
Let \(e_1=x_ix_{i+1}\) and \(e_2=x_jx_{j+1}\) for some \(i \equiv 1\pmod 4\), \(j \equiv 3\pmod 4\), and \(i,j\in [1,\ell]\). Then \(f(e_1) =|m-4s-i|\) and \(f(e_2) =m-j\). If \(f(e_1) =f(e_2)\), then we have either \(j=4s+i\) or \(2m=4s+i+j\). In the former case, we get a contradiction to the fact that \(i \equiv 1\pmod 4\) and \(j\equiv 3\pmod 4\). In the latter case, since \(m=2s+\ell\), we get \(i+j=2\ell\), a contradiction to \(i,j\in[1,\ell -1]\).
Combining Claims 3 and 4, we know that \(f\) is a graceful labeling of \(S\) when \(\ell\) is even. 0◻
The proof of Theorem 4 is now completed. ◻
References↩︎
[1]
A. Rosa, “On certain valuations of the vertices of a graph,” Theory of Graphs (Internat. Sympos., Rome, 1966),
pp. 349–355, 1967.
[2]
S. W. Golomb, “How to number a graph,” Graph theory and computing, pp. 23–37, 1972.
[3]
G. Ringel, Theory of graphs and its applications. 1964.
[4]
C. Huang, A. Kotzig, and A. Rosa, “Further results on tree labelings,” Utilitas Mathematica, vol. 21, pp. 31–48, 1982.
[5]
P. Hrncíar and A. Haviar, “All trees of diameter five are graceful,” Discrete Mathematics, vol. 233, pp. 133–150, 2001.
[6]
J. A. Gallian, 28th edition, October 30, 2025“A dynamic survey of graph labeling,” Electronic Journal of Combinatorics, vol. DS6, 2025, [Online]. Available: https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.
[7]
P. Bahls, S. Lake, and A. Wertheim, “Gracefulness of families of spiders,” Involve, vol. 3, no. 3, pp. 241–247, 2010, doi: 10.2140/involve.2010.3.241.
[8]
A. Panpa and T. Poomsa-ard, “On graceful spider graphs with at most four legs of lengths greater than one,” Journal of Applied Mathematics, vol. 3, pp. 1–5, 2016,
doi: 10.1155/2016/5327026.
[9]
A. Panpa, S. Imnang, and T. Wasuanankul, “Graceful labeling of spider graphs with at most five legs,” Journal of Applied Mathematics, p. 5, 2025.
[10]
R. Cattell, “Graceful labellings of paths,” Discrete Math., vol. 307, no. 24, pp. 3161–3176, Nov. 2007, doi: 10.1016/j.disc.2007.03.046.
[11]
K. Saengsura and T. Poomsa-ard, “Graceful labeling of some spider graphs,” European Journal of Pure and Applied Mathematics, vol. 18, no. 2, p. 5305, 2025.
[12]
A. Rosa, “Labeling snakes,” Ars Combinatoria, vol. 3, pp. 67–74, 1977.