International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 1801925, 8 pages

https://doi.org/10.1155/2019/1801925

## On Edge Magic Total Labeling of (7, 3)-Cycle Books

Department Mathematics, Bengkulu University, Jalan W.R.Supratman, Kandanglimun, 38371, Indonesia

Correspondence should be addressed to Mudin Simanihuruk; ua.moc.oohay@kuruhinamisnidum

Received 13 September 2018; Revised 24 November 2018; Accepted 2 December 2018; Published 2 January 2019

Academic Editor: Dalibor Froncek

Copyright © 2019 Baki Swita et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A graph* G *is called (*a, b*)-*cycle books B*[(*C*_{a}*, m*)*, *(*C*_{b}*, n*)*, P*_{t}] if* G *consists of* m *cycles* C*_{a} and* n *cycles* C*_{b} with a common path* P*_{t}. In this article we show that the (7*, *3)-cycle books* B*[(*C*_{7}*, *1)*, *(*C*_{3}*, n*)*, P*_{2}] admits edge-magic total labeling. In addition we prove that (*a, *3)-cycle books* B*[(*C*_{a}*, *1)*, *(*C*_{3}*, n*)*, P*_{2}] admits super edge-magic total labeling for* a* = 7 and extends the values of* a *to 4*x *− 1 for any positive integer* x*. Moreover we prove that the (7, 3)-cycle books* B*[(*C*_{7}*, *2)*, *(*C*_{3}*, n*)*, P*_{2}] 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 C_{a} and C_{b} be the cycles of order a and b, respectively. A graph G is called (a, b)-*cycle books *B[(C_{a}, m), (C_{b}, n), P_{t}] if G consists of m cycles C_{a} and n cycles C_{b} with a common path P_{t}.

From now on the graphs (a, b)-*cycle books* B[(C_{a}, m), (C_{b}, n), P_{t}] 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 B_{r} [2]. If a = b, m + n = r, and t = 2, then the graphs B(a, m, b, n, t) are the graphs cycle books B_{a,r} [1] and it is the graphs cycle books Θ(C_{a})^{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 P_{2r}(+)N_{m+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)} [+] N_{n} ([9], p.7) is isomorphic to B((2x + 1)(2y + 1), 1, 4, n, 3) and P_{2x}(+)N_{n} [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)} [+] N_{n} ([9], Theorem 4.13.1,p.93) from the super edge-magic total labeling of P_{2x}(+)N_{n}. 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[(C_{a}, 1), (C_{3}, n), P_{2}] 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[(C_{7}, 2), (C_{3}, n), P_{2}] 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.*

*∪*