Abstract
A graph G is called (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] if G consists of m cycles Ca and n cycles Cb with a common path Pt. In this article we show that the (7, 3)-cycle books B[(C7, 1), (C3, n), P2] admits edge-magic total labeling. In addition we prove that (a, 3)-cycle books B[(Ca, 1), (C3, n), P2] admits super edge-magic total labeling for a = 7 and extends the values of a to 4x − 1 for any positive integer x. Moreover we prove that the (7, 3)-cycle books B[(C7, 2), (C3, n), P2] admits super edge-magic total labeling.
1. Introduction
Let G be a graph such that V (G) = p and E(G) = q. An edge-magic total labeling of G is a bijective function f: V (G)∪E(G) → , 2,..., p+ such that f (w) + f (wz) + f (z) = k for any edge wz ∈ E(G). Moreover G is said to be an edge-magic total. If f (V (G)) = , 2,..., , then f is the super edge-magic total labeling of G and G is said to be a super edge-magic total.
In this paper we generalize the definition of cycle books [1] and we investigate their edge-magic total labeling. Let m, n, a, b, and t be any positive integers and let Ca and Cb be the cycles of order a and b, respectively. A graph G is called (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] if G consists of m cycles Ca and n cycles Cb with a common path Pt.
From now on the graphs (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] is denoted by B(a, m, b, n, t). If a = b = 4, m + n = r, and t = 2, then the graphs B(4, m, 4, n, t) are the graphs cycle books Br [2]. If a = b, m + n = r, and t = 2, then the graphs B(a, m, b, n, t) are the graphs cycle books Ba,r [1] and it is the graphs cycle books Θ(Ca)r [3]. If m = n = 1 and t = 2, then the graphs B(a,m,b,n,t) are a cycle with a chord [4]. If a = b = 2r + 1, m + n ≥ 2, and t = 2r, then the graphs B(a, m, b, n, t) are called a graph with many odd cycles P2r(+)Nm+n [5].
It is an open problem to determine the edge-magic total labeling of B(a, m, b, n, t) (see Research Problem 2.7, p.39 [1] for the case B(a, m, a, n, 2)). However some authors provided a partial solution to this problem. Figueroa-Centeno, Ichishima, and Muntaner-Batle [2] proved that B(4, m, 4, n, 2) is an edge-magic total. Moreover they proved that B(4, m, 4, n, 2) is not super edge-magic total for m + n ≡ 1, 3, 7 mod and m + n = 4, B(4, m, 4, n, 2) is super edge-magic total for m + n = 2, 5, 6, 8, 10, 11 and conjecture that B(4, m, 4, n, 2) is super edge-magic total if and only if m + n is even or m + n ≡ 5(mod 8). Gallian [6] pointed out that Yuansheng et al. [7] proved this conjecture when m + n is even. Later on Yegnanarayanan and Vaidhyanathan [8] proved that B(4, m, 4, n, 2) is an edge-magic total.
Note that C(2x+1)(2y+1) [+] Nn ([9], p.7) is isomorphic to B((2x + 1)(2y + 1), 1, 4, n, 3) and P2x(+)Nn [5] is isomorphic to B(2x+1, μ, 2x+1, ν, 2x), μ+ν = n. Singgih provided a new method to construct super edge-magic total labeling of graph C(2x+1)(2y+1) [+] Nn ([9], Theorem 4.13.1,p.93) from the super edge-magic total labeling of P2x(+)Nn. Hence super edge-magic total labeling of B((2x+1)(2y+1), 1, 4, n, 3) is deduced from the super edge-magic total labeling of B(2x + 1, μ, 2x + 1, ν, 2x), μ + ν = n.
This paper is organized as follows. In Section 3.1 we prove that the graphs B(7, m, 3, n, 2) are an edge-magic total for any integer n ≥ 1 and m = 1. In Section 3.2 we prove that B[(Ca, 1), (C3, n), P2] admits super edge-magic total for a = 7 and extends the values of a to 4x − 1 for any positive integers x. Moreover we prove that the graphs B[(C7, 2), (C3, n), P2] admit super edge- magic total. The results of this paper are developed from [10, 11].
2. Preliminary Notes
In this section we provide some previous results on super edge-magic total labeling of a graph. Figueroa-Centeno, Ichishima, and Muntaner-Batle [2] proved some necessary conditions for super edge-magic total labeling of a graph. We need them to prove the main results of this paper.
Theorem 1 (see [2]). A -graph G is super edge-magic if and only if there exists a bijective function such that the set S = w + fz: wz EG exists of q consecutive integers. In such a case, f extends to a super edge-magic labeling of G with magic constant k = p + q + s, where s = and S = – p+1, k – p + 2,...., k – p + q.
Theorem 2 (see [2]). Let G be super edge-magic labeling -graph and f be a super edge-magic labeling of G. Then In particular .
3. Main Results
We sometimes interchange the role of vw and (v, w) as an element of E(G). From now on we assume that B(a, m, b, n, 2) is defined as follows. Let G be the graphs B(a, m, b, n, 2). Let A = , , B = , ,..., , , ,..., ,..., , ,..., , and D = , ,..., , , ,..., ,..., , ,..., . We define the vertex set V (G) = A ∪ B ∪ D and the edge set E(G) = ∪, ), 3 ≤ i ≤ a-1, 1 ≤ j ≤ ∪, ), 3 ≤ i ≤ b-1, 1 ≤ j ≤ ∪, ), (, ), 1 ≤ i ≤ m, 1 ≤ i ≤ ∪, ), (, ), 1 ≤ i ≤ m, 1 ≤ j ≤ . A graph B(a, m, b, n, 2) is shown in Figure 1.
3.1. Edge Magic Total Labeling of (7,3)-Cycle Books
In this section we provide an edge-magic total labeling of (7, 3)-cycle books B(7, 1, 3, n, 2).
Theorem 3. Let G be a 7, 3-cycle books B7, 1, 3, n, 2. Then G is an edge-magic with magic constant k = 4n + 20.
Proof. Let G be a (7,3)-cycle books B(7, 1, 3, n, 2) in Figure 1 with = p and = q. We first notice that p = n + 7 and q = 2n + 7. We define the bijective function f: V (G) ∪ E(G) → , 2,..., p + or f: V (G) ∪ E(G) → , 2,..., 3n + 1 as in (3):It is easy to verify that fw + fwz + fz = 4n + 20 for any edge , hence the theorem.
3.2. Super Edge Magic Total Labeling of (7,3)-Cycle Books
In this section we prove that the graphs B(a, 1, 3, n, 2) are super edge-magic total for a = 4x−1, x ≥ 2, x is integer, and the graphs B(7, 2, 3, n, 2) are super edge-magic total for any positive integer n. First we provide some lemmas.
Lemma 4. Let G be the graphs Ba, m, b, n, 2 in Figure 1, = p, = q, and let f be a super edge-magic total labeling of G.
Let S = (w) + f (z): f or any edge wz ∈ EG be q consecutive integers and s=. Then
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1. Let V(G) = p and E(G) = q. We first notice that deg() = deg() = m + n + 1, deg() = deg() =... = deg() = 2,..., deg() = deg() =... = deg() = 2, deg() = deg() =... = deg() = 2,..., deg() = deg() =... = deg() = 2.
By Theorem 2 we have (m+n+1)[+] + = + . Hence (m+n-1)[+] + = + . Moreover (m+n-1)[+] + 2(1 + 2 +...+ ) = + . From the last equation we conclude that hence the lemma.
Lemma 5. Let G be the graphs Ba, m, b, n, 2 in Figure 1 with = p, = q, a = 7, and b = 3 and let f be a super edge-magic total labeling of G. Let S = w+f z: f or any edge wz ∈ EG be q consecutive integers and s and s = . Then
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p, = q, a = 7, and b = 3. Let = p and = q. We first prove the upper bound of s. Notice that f () + f () ≤ p + p – 1. By Lemma 4 and the last inequality we have hence the upper bound of s. Next we prove the lower bound of s. Notice that f () + f () ≥ 3. By Lemma 4 and the last inequality we have s ≥ - , hence the lemma.
Lemma 6. Let G be the graphs Ba, m, b, n, 2 in Figure 1 with = p, = q, a = 7, b = 3, m = 1, and n = 1 and let f be an edge-magic total labeling of G. Let S = w + f z: f or any edge wz ∈ EG be q consecutive integers and s = . Then G is a super edge-magic total if and only if(i)f + f =9s – 36,(ii)3 ≤ s ≤ 5,(iii)k = p + q + s.
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p, = q, a = 7, b = 3, m = 1, and n = 1 and let G be a super edge-magic total.
Note that p = (a−2)m+2+(b−2)n and q = (a−3)+2m+1+(b−3)n+2n. Let f be an edge-magic total labeling of G. If f is a super edge-magic total labeling of G, the conditions (i) and (ii) follow from (4) and (6) of Lemmas 4 and 5, respectively, and the condition (iii) follows from Theorem 1.
Let f satisfy the conditions (i), (ii), and (iii). We will prove that G is super edge-magic total for a = 7, b = 3, m = 1, and n = 1. We first notice that p = 8 and q = 9.
Claim 1 (s = 5). If s = 4, then f () + f () = 0, a contradiction to f () + f () ≥ 3, hence s = 5 and the claim.
By (iii) and Claim 1, we have k = 22 and by (i) we have f () + f () = 9. Hence either f () = 1 and f () = 8, f () = 2 and f () = 7, f () = 3 and f () = 6, or f () = 4 and f () = 5.
Case 1 (f () = 1 and f () = 8). Using k = 22, we define the bijection f1(x) in (8) such that f1() = f () = 1 and f1() = f () = 8.The function f1(x) is depicted in Figure 2.
Using the definition of f1(x) we get the following: f1 () +f () = 5, f1 () + f1 () = 6, f1 () + f1 () = 7, f1 () + f1 () = 8, f1 () + f1 () = 9, f1 () + f1 () = 10, f1 () + f1 () = 11, f1 () + f1 () = 12, and f1 () + f1 () = 13. Hence S = (u) + f1(w): uw ∈ E( = , 6,..., consists of q = 9 consecutive integers. Thus by Theorem 1 and Claim 1, we conclude that G is super edge-magic total. Moreover we conclude f1(uv) =k − (f1(u) + f1(v)) for all uv ∈ E(G).
Case 2 (f () = 2 and f () = 7). Using k = 22, we define the bijection f2(x) in (9) such that f2() = f () = 2 and f2() = f () = 7.Using the definition of f2(x) we get the following: f2() + f2() = 5, f2 () + f2() = 6, f2 () + f2 () = 7, f2 () + f2 () = 8, f2 () + f2 () = 9, f2 ()+f2 () = 10, f2 ()+ f2 () = 11, f2()+f2 () = 12, and f2 ()+f2 () = 13. Hence S = (u) + f2(w): uw ∈ E( = , 6,..., consists of q = 9 consecutive integers. Thus by Theorem 1 and Claim 1, we conclude that G is super edge-magic total. Moreover we conclude f2(uv) = k − (f2(u) + f2(v)) for all uv ∈ E(G) and it is shown in Figure 3.
Case 3 (f () = 3 and f () = 6). Using k = 22, we define the bijection f3(x) in (10) such that f3() = f () = 3 and f3() = f () = 6.The function f3(x) is depicted in Figure 4. Using the definition of f3(x) we get the following: f3() + f3() = 5, f3 () + f3 () = 6, f3 () + f3 () = 7, f3 () + f3 () = 8, f3 () + f3 () = 9, f3 () + f3 () = 10, f3 () + f3 () = 11, f3 () + f3 () = 12, and f3 () + f3 () = 13. Hence S = (u) + f3(w): uw ∈ E( = , 6,..., consists of q = 9 consecutive integers. Thus by Theorem 1 and Claim 1, we conclude that G is super edge-magic total. Moreover we conclude f3(uv) = k − (f3(u) + f3(v)) for all uv ∈ E(G).
Case 4 (f() = 4 and f() = 5). Using k = 22, we define the bijection f4(x) in (11) such that f4() = f () = 4 and f4() = f () = 5.The function f4(x) is depicted in Figure 5. Using the definition of f4(x) we get the following: f4 () + f4 () = 5, f4 () + f4 () = 6, f4 () + f4 () = 7, f4 () + f4 () = 8, f4 () + f4 () = 9, f4 () + f4 () = 10, f4 () + f4 () = 11, f4 () + f4 () = 12, and f4 () + f4 () = 13. Hence S = (u) + f4(w): uw ∈ E( = , 6,..., 1 consists of q = 9 consecutive integers. Thus by Theorem 1 and Claim 1, we conclude that G is super edge-magic total. Moreover we conclude f4(uv) = k − (f4(u) + f4(v)) for all uv ∈ E(G), hence the lemma.
Now we are ready to state our main result.
Theorem 7. Let G be the graphs Ba, m, b, n, 2 in Figure 1 with = p, = q, a = 7, m = 1, and b = 3. Then G is a super edge-magic total. Moreover there are two bijective functions g1V G and g2V G with the magic constant k = 3n + 19.
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p, E(G) = q, a = 7, m = 1, and b = 3. We first notice that p = (a − 2)m + (b − 2)n + 2 and q = (a − 3)m + 2m + 1 + (b − 3)n + 2n. For a = 7, m = 1, and b = 3 and n = 1 we have p = 8 and q = 9. Using the definition of the bijective function f1(x) in Case 1 of Lemma 6 together with s = 5 in Claim 1 of Lemma 6, we define the following bijective function g1(x) in (12):The function g1(x) is depicted in Figure 6. Using the definition of g1(x) we get the following: g1() + g1() = 5, g1() + g1() = 6, g1() + g1() = i + 5, i ∈, 3,..., , g1() + g1() = n + 6, g1()+g() = n+7, g1() + g1() = n+8, g1()+g() = n+9, g1()+g1() = n + 10, g1() + g1() = n + 11, g1() + g1 () = n + 11 + i, i ∈, 2, 3,..., . Hence S = (u) + g1(w): uw ∈ E( = , 6,..., i + 5, n + 7, n +8, n+9, n+10, n+11, n+i+11, i ∈,..., consists of q = 2n+7 consecutive integers. Thus by Theorem 1 and Claim 1 of Lemma 6, we conclude that G is super edge-magic total with magic constant k = p+q+s = n+7+2n+7+5 = 3n + 19. Moreover, by Theorem 1, we conclude g1(uv) = k − (g1(u) + g1(v)) for all uv ∈ E(G).
Using the definition of the bijective function f2(x) in Case 2 of Lemma 6 together with s = 5 in Claim 1 of Lemma 6, we define the following bijective function g2(x) in (13):The function g2(x) is depicted in Figure 7. Using the definitions of g2(x) we get the following: g2() + g2() = i +4, i ∈, 2,..., , g2() + g2() = n + 5, g2() + g2() = n + 6, g2() + g2() = n+7, g2()+g2() = n+8, g2()+g2() = n+8+i, i ∈, 2,..., , g2() + g2() = 2n + 9, g2() + g2() = 2n + 10, g2() + g2() = 2n + 11. Hence S = (u) + g2(w): uw ∈ E( = , 6,..., i + 4, n + 5, n + 6, n + 7, n + 8, n + 8 + i, 2n + 9, 2n + 10, 2n + 11, i ∈, 2,..., consists of q = 2n + 7 consecutive integers. Thus by Theorem 1 and Claim 1 of Lemma 6, we conclude that G is super edge-magic total with magic constant k = p + q + s = n + 7 + 2n + 7 + 5 = 3n + 19.
Moreover, by Theorem 1, we conclude g2(uv) = k − (g2(u) + g2(v)) for all uv ∈ E(G), hence the theorem.
Theorem 8. Let G be the graphs Ba, m, b, n, 2 in Figure 1 such that = p, = q, a = 4x − 1, m = 1, and b = 3.
Let α2: V G → , 2,..., be a bijective function such thatThen α2v is a super edge-magic total labeling of G with the magic constant k = 10x + 3n − 1, v ∈ V G.
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p and = q. Let a = 4x − 1, m = 1, and b = 3. Notice that p = 4x − 1 + n and q = 2n + 4x − 1. Let S = (u) + α2(w): uw ∈ E(. We show that S consists of q consecutive integers.
Using the definition of α2(v) in (14), v ∈ V (G), we get the following:Hence by (15) we have S = x + 1, 2x + 2, 2x + 3,..., 2x + n + 1, 2x + n + 2,..., 3x + n −1, 3x + n, 3x + n + 1,..., 4x + n − 1, 4x + n, 4x + n + 1,..., 5x + n − 1, 5x + n, 5x +n + 1,..., 6x + n − 1, 6x + n, 6x + n + 1,..., 6x + 2n − 2, 6x + 2n − consists of q consecutive integers. Thus by Theorem 1 we conclude that G is super edge-magic total. Notice that s = 2x + 1. Hence k = p + q + s = 10x + 3n − 1. Moreover, by Theorem 1, we conclude α2(uv) = k − (α2(u) + α2(v)) for all uv ∈ E(G), hence the theorem.
Lemma 9. Let G be the graphs Ba, m, b, n, 2 in Figure 1 with = p, = q, a = 7, m = 2, b = 3, and n = 1 and let f be an edge-magic total labeling of G. Let S = w + f z: f or any edge wz ∈ EG be q consecutive integers and s = . Then G is a super edge-magic total if and only if(i)+=,(ii)6 ≤ s ≤ 7,(iii)k = p + q + s.
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p, = q, a = 7, m = 2, b = 3, and n = 1 and let G be a super edge-magic total. Let f be an edge-magic total labeling of G. If f is a super edge-magic total labeling of G, the conditions (i) and (ii) follow from Lemmas 4 and 5, respectively, and the condition (iii) follows from Theorem 1.
Let f satisfy the conditions (i), (ii), and (iii). We first notice that p = (a − 2)m + (b − 2)n + 2 and q = (a − 3)m + 2m + 1 + (b − 3)n + 2n. We will prove that G is super edge-magic total for a = 7, m = 2, b = 3, and n = 1. Notice that p = 13 and q = 15.
Claim 1 (s = 7). If s = 6, then += 6,5 (by (i)); a contradiction to + is an integer, hence s = 7 and the claim. By (iii) and Claim 1, we have k =35 and by (i) we have + = 14.
Next we will show that there is a bijection h1(x) such that = , =, + = 14, and h1(x) is a super edge-magic total labeling of B(7, 2, 3, 1). Let = = 2 and ==12. Using k = 35, we define the bijection h1(x) in (16):Using the definition of h1(x) we get the following: h1() + h1() = 7, h1()+h1() = 8, h1()+h1() = 9, h1()+h1() = 10, h1()+h1() = 11, h1() + h1() = 12, h1() + h1() = 13, h1() + h1() = 14, h1() + h1() = 15, h1() + h1() = 16, h1() + h1() = 17, h1() + h1() = 18, h1() + h1() = 19, h1() + h1() = 20, h1() + h1() = 21. Hence S = (u) + h1(w): uw ∈ E( = , 8,..., consists of q = 15 consecutive integers. Thus by Theorem 1 and Claim 1, we conclude that G is super edge-magic total. Moreover we conclude h1(uv) = k − (h1(u) + h1(v)) for all uv ∈ E(G) and it is shown in Figure 8, hence the theorem.
Theorem 10. Let G be the graphs Ba, m, b, n, 2 in Figure 1 with = p and = q, a = 7, m = 2, and b = 3. Then G is a super edge-magic total with the magic constant k = 3n + 32.
Proof. Let G be the graphs B(a, m, b, n, 2) in Figure 1 with = p, = q, a = 7, m = 2, and b = 3. We first notice that p = (a − 2)m + (b − 2)n + 2 and q = (a − 3)m + 2m + 1 + (b − 3)n + 2n. Using the definition of the bijective function h1(x) and s = 7 in Lemma 9, we define the following bijective function h2(x) in (17):The function h2(x) is depicted in Figure 9. Using the definition of h2(x) we get the following: h2()+h2() = i + 6, i ∈,..., , h2() + h2() = n + 7, h2()+h2() = n+8, h2()+h2() = n+9, h2() + h2() = n + 10, h2() + h2() = n + 11, h2() + h2() = n + 12, h2() + h2() = n + 13, h2() + h2() = n + 14, h2() + h2() = n + 15, h2() + h2() = n + 15 + i, i ∈,..., , h2() + h2() = 2n + 16, h2() +h2() = 2n + 17, h2() + h2() = 2n + 18, h2() + h2() = 2n + 19. Hence S = (u) + h2(w): uw ∈ E( = , 8,..., n + 6, n + 7, n + 8, n + 9, n +10, n + 11, n + 12, n + 13, n + 14, n + 15, n + 16,..., 2n + 15, 2n + 16, 2n + 17, 2n +18, 2n + consists of q = 2n + 13 consecutive integers. Thus by Theorem 1 and Claim 1 of Lemma 9, we conclude that G is super edge-magic total with magic constant k = p + q + s = n + 12 + 2n + 13 + 7 = 3n + 32. Moreover, by Theorem 1, we conclude h2(uv) = k − (h2(u) + h2(v)) for all uv ∈ E(G), hence the theorem.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors thank the Department of Mathematics, Bengkulu University, Indonesia, under research Grant no. 5742.002.051/2017.