1 Introduction
All graphs considered here are finite, undirected and simple. As usual, the vertex set and edge set will be denoted V and respectively. The symbol (G) E(G), A will be denote the cardinality of the set A. Other terminologies or notations not defined here can be found in [2,7,15].
Edge-magic total labelings were introduced by Kotzig and Rosa [8] as follow. An edge-magic total labeling on G is a bijection f from V (G) ∪ E(G) onto {1,2,",|V (G)| + | E(G)|} with the property that, given any edge uv,
\[f(u) + f(v) + f(uv) = k\] for some constan k. It will be convenient to call f (u) + f (v) + f (uv) the edge sum of uv and k the magic sum of f. A graph is called edge-magic total if it admits any edge-magic total labeling.
∗ This work was supported by Hibah Bersaing XII DP3M-DIKTI, DIP Number : 004/XXIII/1/--/2004.
Kotzig and Rosa [9] showed that no complete graph \(K_n\) with n > 6 is edge-magic total and neither is \(K_4\), and edge-magic total labelings for \(K_3, K_5\) and \(K_6\) for all feasible values of k, are described in [14].
In [8] it is proved that every cycle \(C_n\), every caterpillar (a graph derived from a path by hanging any number of pendant vertices from vertices of the path) and every complete bipartite graph \(K_{m,n}\) (for any m and n) are edge-magic total.
Wallis et.al. [14] showed that all paths \(P_n\) and all n-suns (a cycle \(C_n\) with an additional edge terminating in a vertex of degree 1 attached to each vertex of the cycle) are edge-magic total. It was shown in [16] that the Cartesian product \(C_n \times P_m\) admits an edge-magic total labeling for odd n.
It is conjectured that all trees are edge-magic total [8] and all wheels \(W_n\) are edge-magic total whenever \(n \ 3 \ (\text{mod } 4)\) [4]. Enomoto et.al. [4] showed that the conjectures are true for all trees with less than 16 vertices and wheels \(W_n\) for \(n \le 30\). Philips et.al. [12] solved the conjecture partially by showing that a wheel \(W_n\), \(n \equiv 0\) or 1 (mod 4), is edge-magic total. Slamin et.al [13] showed that for \(n \equiv 6 \ (\text{mod } 8)\) every wheel \(W_n\) has an edge-magic total labeling.
An edge-magic total labeling f is called super edge-magic total if \(f(V(G)) = \{1, 2, \dots, |V(G)|\}\) and \(f(E(G)) = \{|V(G)| + 1, |V(G)| + 2, \dots, |V(G)| + |E(G)|\}\). Enomoto et.al. [4] proved that the complete bipartite graphs \(K_{m,n}\) is super edge-magic total if and only if m = 1 or n = 1. They also proved the complete graphs \(K_n\) is super edge-magic if and only if n = 1, 2 or 3.
In this paper we will construct the super edge-magic total graphs by hanging any number of pendant vertices from vertices of the cycles, generalized prisms and generalized Petersen graphs.
2 Results
For \(n \ge 3\) and \(p \ge 1\) we denote by \(C_n + A_p\) a graph which is obtained by adding p vertices and p edges to one vertex of cycles \(C_n\) (say \(v_1\)). The vertex set and the edge set of \(C_n + A_p\) are \(V(C_n + A_p) = \{v_i : 1 \le i \le n\} \cup \{u_j : 1 \le j \le p\}\) and \(E(C_n + A_p) = \{v_i v_{i+1} : 1 \le i \le n-1\} \cup \{v_n v_1\} \cup \{v_1 u_j : 1 \le j \le p\}\).
Let (n,p)-sun be a graph derived from a cycle \(C_n\), \(n \ge 3\), by hanging p pendant vertices from all vertices of the cycle. Let us denote the vertex set of (n,p)-sun by \(V((n,p)-sun)=\{v_i:1\le i\le n\}\cup\{u_{i,j}:1\le i\le n,1\le j\le p\}\) and the edge set by \(E((n,p)-sun)=\{v_iv_{i+1}:1\le i\le n-1\}\cup\{v_nv_1\}\cup\{v_iu_{i,j}:1\le i\le n,1\le j\le p\}\). Observe that |V((n,p)-sun)|=|E((n,p)-sun)|=n(p+1). The cycle \(C_n\), \(n\ge 3\), is super edge-magic total if and only if n is odd (see [4]). Now, we shall investigate super edge-magic total labelings for graphs of \(C_n+A_p\) and (n,p)-sun which are expanded from a cycle \(C_n\).
Define a vertex labeling \(f_1\) and an edge labeling \(f_2\) of \(C_n + A_p\) as follows,
\[f_1(v_i) = \begin{cases} \frac{n+i}{2} & \text{if } i \text{ is odd,} \\ \frac{i}{2} & \text{if } i \text{ is even,} \end{cases}\] \[f_1(u_j) = n+j & \text{for } 1 \le j \le p,\] \[f_2(v_i v_{i+1}) = 2(n+p)+1-i & \text{for } 1 \le i \le n-1,\] \[f_2(v_n v_1) = n+2p+1,\] \[f_2(v_1 u_i) = n+2p+1-j & \text{for } 1 \le j \le p.\]
Theorem 1. If n is odd, \(n \ge 3\) and \(p \ge 1\), then graph \(C_n + A_p\) is super edgemagic total.
Proof. It is easy to verify that the values of \(f_1\) are \(1,2,\dots,n+p\) and the values of \(f_2\) are \(n+p+1,n+p+2,\dots,2n+2p\) and furthermore the common edge sum is \(k=2p+\frac{5n+3}{2}\).
Theorem 2. If n is odd, \(n \ge 3\) and \(p \ge 1\), then graph (n, p) – sun is super edge-magic total.
Proof. Label the vertices and the edges of (n, p) – sun in the following way.
\[\begin{split} f_3(v_i) &= f_1(v_i) \quad \text{for} \quad 1 \leq i \leq n, \\ f_3(u_{1,j}) &= nj+1 \quad \text{for} \ 1 \leq j \leq p, \\ f_3(u_{i,j}) &= n(j+1)+2-i \quad \text{for} \ 2 \leq i \leq n \ \text{and} \ 1 \leq j \leq p, \\ f_4(v_i v_{i+1}) &= 2n(p+1)+1-i \quad \text{for} \ 1 \leq i \leq n, \end{split}\]
\[f_4(v_n v_1) = 2np + n + 1,\] \[f_4(v_i u_{i,j}) = \begin{cases} 2n(p+1) - nj & \text{if } i = 1 \text{ and } 1 \le j \le p, \\ 2np + n(1-j) + \frac{i-1}{2} & \text{if } i \text{ is odd, } 3 \le i \le n \text{ and } 1 \le j \le p, \\ 2n(p+1) - nj + \frac{i-n-1}{2} & \text{if } i \text{ is even, } 2 \le i \le n-1 \text{ and } 1 \le j \le p. \end{cases}\]
We can see that the vertices of (n, p)-sun are labeled by values \(1, 2, \dots\), n(p+1) and the edges are labeled by n(p+1)+1, n(p+1)+2, \(\dots\), 2n(p+1). Furthermore, all edges have the same magic number \(k = 2n(p+1) + \frac{n+3}{2}\).
A generalized Petersen graph P(n,m), \(n \ge 3\) and \(1 \le m \le \lfloor \frac{n-1}{2} \rfloor\), consists of an outer n-cycle \(v_1, v_2, \cdots, v_n\) a set of n spokes \(v_i z_i\), \(1 \le i \le n\), and inner edges \(z_i z_{i+m}\), \(1 \le i \le n\), with indices taken modulo n.
For \(n \ge 5\), m = 2 and \(p \ge 1\), we denote by \(P(n,2) + A_p\) for a graph which is obtained by adding p vertices and p edges to one vertex of P(n,2), say \(v_1\). Hence, \(V(P(n,2) + A_p) = V(P(n,2)) \cup \{u_j : 1 \le j \le p\}\) and \(E(P(n,2) + A_p) = E(P(n,2)) \cup \{v_1u_j : 1 \le j \le p\}\).
Let P(n,2,p) be a graph derived from P(n,2), \(n \ge 5\), by hanging p pendant vertices from all vertices \(v_i\), \(1 \le i \le n\) of P(n,2). Then the vertex set of P(n,2,p) is \(V(P(n,2,p)) = V(P(n,2)) \cup \{u_{i,j} : 1 \le i \le n, 1 \le j \le p\}\) and the edge set is \(E(P(n,2,p)) = E(P(n,2)) \cup \{v_i u_{i,j} : 1 \le i \le n, 1 \le j \le p\}\).
In [11] it is proved that generalized Petersen graphs P(n,2) are edge-magic total. Fukuchi [6] showed that P(n,2) are super edge-magic total.
Theorem 3. If n is odd, \(n \ge 5\) and \(p \ge 1\), then the graph \(P(n,2) + A_p\) has a super edge-magic total labeling.
Proof. Consider a bijection, \(f_5: V(P(n,2) + A_p) \rightarrow \{1, 2, \dots, 2n + p\}\) where,
\[f_5(v_i) = \begin{cases} n + \frac{i}{2} & \text{if } i \text{ is even, } 2 \le i \le n - 1, \\ \frac{3n + i}{2} & \text{if } i \text{ is odd, } 1 \le i \le n, \end{cases}\]
\[f_5(z_i) = \begin{cases} \frac{n-i+4}{4} & \text{if } i \equiv 1 \pmod{4}, \\ \frac{2n-i+4}{4} & \text{if } i \equiv 2 \pmod{4}, \\ \frac{3n-i+4}{4} & \text{if } i \equiv 3 \pmod{4}, \\ \frac{4n-i+4}{4} & \text{if } i \equiv 0 \pmod{4}, \end{cases}\] \[f_5(u_i) = 2n+j \quad \text{for } 1 \le j \le p.\]
We can observe that under the labeling \(f_5\), \(\{f_5(v_i) + f_5(v_{i+1}) : 1 \le i \le n\} = \{\frac{5n+1}{2} + i : 1 \le i \le n\}\) and \(\{f_5(z_i) + f_5(z_{i+2}) : 1 \le i \le n\} = \{\frac{n+1}{2} + i : 1 \le i \le n\}\) with indices taken modulo n. Moreover, \(\{f_5(v_i) + f_5(z_i) : 1 \le i \le n\} = \{\frac{3n+1}{2} + i : 1 \le i \le n\}\) and \(\{f_5(v_1) + f_5(u_j) : 1 \le j \le p\} = \{\frac{7n+1}{2} + j : 1 \le j \le p\}\). The elements of the set \(\{f_5(v_i) + f_5(v_{i+1}) : 1 \le i \le n\} \cup \{f_5(z_i) + f_5(z_{i+2}) : 1 \le i \le n\} \cup \{f_5(v_i) + f_5(z_i) : 1 \le i \le n\} \cup \{f_5(v_1) + f_5(u_j) : 1 \le j \le p\}\) form an arithmetic sequence \(\frac{n+1}{2} + 1\), \(\frac{n+1}{2} + 2\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} + 1\), \(\frac{7n+1}{2} +\)
Theorem 4. If n is odd, \(n \ge 5\) and \(p \ge 1\), then the graph P(n,2,p) has a super edge-magic total labeling.
Proof. Define a bijection, \(f_6: V(P(n,2,p)) \rightarrow \{1,2,\dots,n(p+2)\}\) as follows,
\[f_6(v_i) = f_5(v_i)\] and \(f_6(z_i) = f_5(z_i)\) for \(1 \le i \le n\),
\(f_6(u_{1,j}) = n(j+1)+1\) for \(1 \le j \le p\),
\(f_6(u_{i,j}) = n(j+2)+2-i\) for \(2 \le i \le n\) and \(1 \le j \le p\).
We can see that under the vertex labeling \(f_6\) the values \(f_6(x)+f_6(y)\) of all edges \(xy \in E(P(n,2,p))\) constitute an arithmetic sequence \(\frac{n+1}{2}+1, \frac{n+1}{2}+2, \cdots, \frac{7n+1}{2}, \frac{7n+1}{2}+1, \cdots, \frac{7n+1}{2}+np\). If we complete the edge labeling with the consecutive values in the set \(\{n(p+2)+1, n(p+2)+2, n(p+2)+3, \cdots, 5n+2np\}\) then we can obtain total labeling where \(f_6(x)+f_6(y)+f_6(xy)=\frac{11n+3}{2}+2np\) for every edge \(xy \in E(P(n,2,p))\).
In the sequel we shall consider a graph of a generalized prism which can be defined as the Cartesian product \(C_n \times P_m\) of a cycle on n vertices with a path on m vertices.
Let \(V(C_n \times P_m) = \{v_{i,k} : 1 \le i \le n \text{ and } 1 \le k \le m\}\) be the vertex set and \(E(C_n \times P_m) = \{v_{i,k}v_{i+1,k} : 1 \le i \le n \text{ and } 1 \le k \le m\} \cup \{v_{i,k}v_{i,k+1} : 1 \le i \le n \text{ and } 1 \le k \le m-1\}\) be the edge set, where i is taken modulo n. For \(n \ge 3\), \(m \ge 2\) and \(p \ge 1\), we will consider a graph \((C_n \times P_m) + A_p\) (respectively a graph \((C_n \times P_m) + \sum_{i=1}^n A_p^i\)) which is obtained by adding p vertices and p edges to one vertex of \(C_n \times P_m\), say \(v_{1,m}\) (respectively to all vertices \(v_{i,m}\), \(1 \le i \le n\) of \(C_n \times P_m\)). Thus \(V((C_n \times P_m) + A_p) = V(C_n \times P_m) \cup \{u_j : 1 \le j \le p\}\),
\[V((C_n \times P_m) + \sum_{i=1}^n A_p^i) = V(C_n \times P_m) \cup \{u_{i,j} : 1 \le i \le n, 1 \le j \le p\},\] \[E((C_n \times P_m) + A_p) = E(C_n \times P_m) \cup \{v_{1,m}u_j : 1 \le j \le p\}, \text{ and}\] \[E((C_n \times P_m) + \sum_{i=1}^n A_p^i) = E(C_n \times P_m) \cup \{v_{i,m}u_{i,j} : 1 \le i \le n, 1 \le j \le p\}.\]
Figueroa-Centeno et.al. [5] showe that the generalized prism \(C_n \times P_m\) is super edge-magic if n is odd and \(m \ge 2\).
The next two theorems show super edge-magic total labelings of graphs \((C_n \times P_m) + A_p\) and \((C_n \times P_m) + \sum_{i=1}^n A_p^i\).
Theorem 5. If n is odd, \(n \ge 3\), \(m \ge 2\) and \(p \ge 1\), then the graph \((C_n \times P_m) + A_p\) has a super edge-magic total labeling.
Proof. If m is even, \(m \ge 2\), \(1 \le k \le m\), \(1 \le i \le n\), then we construct a vertex labeling \(f_7\) in the following way,
\[f_7(v_{i,k}) = \begin{cases} n(k-1) + \frac{i+1}{2} & \text{if } i \text{ is odd and } k \text{ is odd,} \\ nk + \frac{i-n+1}{2} & \text{if } i \text{ is even and } k \text{ is odd,} \\ nk + \frac{i-n}{2} & \text{if } i \text{ is odd and } k \text{ is even,} \\ n(k-1) + \frac{i}{2} & \text{if } i \text{ is even and } k \text{ is even,} \end{cases}\]
\(f_7(u_j) = mn + j\) for \(1 \le j \le p\).
If m is odd, \(m \ge 3\), \(1 \le k \le m\), \(1 \le i \le n\), then we define a vertex labeling \(f_8\) as follows,
\[f_8(v_{i,k}) = \begin{cases} \frac{n+i}{2} + n(k-1) & \text{if } i \text{ is odd and } k \text{ is odd,} \\ \frac{i}{2} + n(k-1) & \text{if } i \text{ is even and } k \text{ is odd,} \\ nk & \text{if } i = 1 \text{ and } k \text{ is even,} \\ n(k-1) + \frac{i-1}{2} & \text{if } i \text{ is odd and } k \text{ is even,} \\ n(k-1) + \frac{n+i-1}{2} & \text{if } i \text{ is even and } k \text{ is even,} \\ f_8(u_i) = mn + j \text{ for } 1 \le j \le p. \end{cases}\]
It is easy to verify that for each edge \(xy \in E((C_n \times P_m) + A_p)\) the values \(f_7(x) + f_7(y)\) and \(f_8(x) + f_8(y)\) form an arithmetic sequence \(\frac{n+1}{2} + 1\), \(\frac{n+1}{2} + 2\), \(\cdots\), \(2mn - \frac{n-1}{2}\), \(2mn - \frac{n-3}{2}\), \(\cdots\), \(2mn - \frac{n-1}{p} + p\).
Let \(f_9\) be a bijection from \(E((C_n \times P_m) + A_p)\) onto \(\{1, 2, \dots, 2nm - n + p\}\). We can combine the vertex labeling \(f_7\) (or \(f_8\)) and the edge labeling \(f_9 + mn + p\) such that the resulting labeling is total and the edge sum for each edge \(xy \in E((C_n \times P_m) + A_p)\) is equal to \(3mn + \frac{3-n}{2} + 2p\).
Theorem 6. If n is odd, \(n \ge 3\), \(m \ge 2\), and \(p \ge 1\), then the graph \((C_n \times P_m) + \sum_{i=1}^n A_p^i\) has a super edge-magic total labeling.
Proof. Define vertex labeling \(f_{10}\) and \(f_{11}\) such that :
\[f_{10}(v_{i,k}) = f_7(v_{i,k})\] if \(m\) is even, \(1 \le k \le m\), \(1 \le i \le n\), \(f_{11}(v_{i,k}) = f_8(v_{i,k})\) if \(m\) is odd, \(1 \le k \le m\), \(1 \le i \le n\), \(f_{10}(u_{1,j}) = f_{11}(u_{1,j}) = n(m+j-1)+1\) for \(1 \le j \le p\),
\[f_{10}(u_{i,j}) = f_{11}(u_{i,j}) = n(m+j) - i + 2\] for \(2 \le i \le n\) and \(1 \le j \le p\).
We can see that vertices of \((C_n \times P_m) + \sum_{i=1}^n A_p^i\) are labeled by values 1, 2, 3,
\(\cdots\), n(m+p) and \(f_t(x)+f_t(y)\) for all edges \(xy \in (C_n \times P_m)+\sum_{i=1}^n A_p^i\) and \(t \in \{10,11\}\) constitute an arithmetic sequence \(\frac{n+1}{2}+1,\frac{n+1}{2}+2,\cdots,2mn-\frac{n-1}{2}+np\).
We can complete the edge labeling of \((C_n \times P_m) + \sum_{i=1}^n A_p^i\) with values in the set \(\{n(m+p)+1, n(m+p)+2, \cdots, n(3m+2p-1)\}\) consecutively such that the common edge sum is \(k = 3mn + 2pn - \frac{n-3}{2}\). Thus the total labeling of \((C_n \times P_m) + \sum_{i=1}^n A_p^i\) is super edge-magic and the theorem is proved.
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