ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS

Transcription

1 Bull. Korean Math. Soc. 5 (008), No., pp. ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Ji Yeon Park, Jin Hyuk Choi, and Jae-Hyeong Bae Reprinted from the Bulletin of the Korean Mathematical Society Vol. 5, No., February 008 c 008 The Korean Mathematical Society

2 Bull. Korean Math. Soc. 5 (008), No., pp. ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Ji Yeon Park, Jin Hyuk Choi, and Jae-Hyeong Bae Abstract. A graph G = (V, E) is called super edge-magic if there exists a one-to-one map λ from V E onto {,,,..., V + E } such that λ(v ) = {,,..., V } and λ(x) + λ(xy) + λ(y) is constant for every edge xy. In this paper, we investigate whether some families of graphs are super edge-magic or not.. Introduction Throughout this paper, we assume that all graphs are finite, simple and undirected. A graph G has vertex set V (G) and edge set E(G) and we let V (G) = ν and E(G) = ϵ. A general reference theoretic notions is West []. Given a graph G, V (G) = V and E(G) = E. A labeling for a graph is a map that takes graph elements to numbers (usually positive or non-negative integers). Various authors have introduced labelings that generalize the idea of magic square. Kotzig and Rosa [] defined a magic labeling to be a total labeling on the vertices and edges in which the labels are the integers from to V (G) + E(G). The sum of labels on an edge and its two endpoints is constant. In Ringel and Llado [] redefined this type of labeling as edge-magic. Also, Enomoto et al [] have introduced the name super edge-magic for magic labelings in the sense of Kotzig and Rosa, with the added property that the ν vertices receive the smaller labels, {,,..., ν}. A one-to-one map λ from V E onto the integers {,,..., ν + ϵ} is an edge-magic labeling if there is a constant k so that for any edge xy, λ(x) + λ(xy) + λ(y) = k. The constant k is called the edge magic number for λ. An edge-magic labeling λ is called super edge-magic if λ(v ) = {,,..., ν} and λ(e) = {ν +, ν +,..., ν + ϵ}. A graph G is called edge-magic (resp. super edge-magic) if there exists an edge-magic (resp. super edge-magic) labeling of G. Received December, 005; Revised September, Mathematics Subject Classification. 05C8, 8R0, 0B8, 0C5. Key words and phrases. edge magic labeling, super edge-magic graphs, magic number. This research was supported by the Kyung Hee University Research Fund in 005.(KHU ). c 008 The Korean Mathematical Society

3 JI YEON PARK, JIN HYUK CHOI, AND JAE-HYEONG BAE Kotzig and Rosa [] and Wallis et al [5] showed the edge-magicness for cycles C n and some variants of C n. Also, Enomoto et al [] proved that a cycle C n is super edge-magic if and only if n is odd. In this paper, we also find super edge-magic variants of C n. First of all, we show that an (n, )-kite is super edge-magic if n is even and that the converse is also true. Finally, we prove that the graph K C n is super edge-magic if n is even. It was motivated by Wallis [] question to characterize edge-magic graphs.. Main results We take some families of graphs to see whether or not they are super edgemagic. An (n, t)-kite is a cycle of length n with a t-edge path (the tail) attached to one vertex. In [5], Wallis et al. showed that (n, )-kite is edge-magic. The following proposition shows that an (n, )-kite is super edge-magic if n is even and that the converse is also true. Proposition.. Let G = (n, )-kite. The graph G is super edge-magic if and only if n is even. Proof. Let v 0, v,..., v n, v 0 be a vertex sequence of C n, a vertex v is adjacent to v 0 and a vertex w is adjacent to v. Suppose there exists a super edge-magic labeling λ of G with the magic number k. Set λ(v 0 ) = α and λ(w) = β. Then k(n + ) = {λ(x) + λ(xy) + λ(y)} xy E(G) = = x V (G) λ(x) + λ(v 0 ) λ(w) + (n + )(5n + ) + α β. xy E(G) λ(xy) This implies that k α β n+ = 5n+ is a rational number that is not integer, since 0 < α β n+ <. Thus n must be an even. Let n = m + for a nonnegative integer m. V (G) = {v 0, v,..., v n, v, w} and E(G) = {v i v i+ 0 i n } {v n v 0 } {v 0 v} {vw}. We define a labeling λ : V E {,,..., m+8} as follows: Case where m is odd. (m i + )/ if i = 0,,..., m, m (m i + )/ if i =,,..., m, m (m i + )/ if i = m +, m +,..., m (5m i + 0)/ if i = m +, m +,..., m +, λ(v) = (m + )/, λ(w) = (m + 5)/,

4 ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Figure. An illustration for the labeling given in the proof of Proposition.. m + i + if 0 i m λ(v i v i+ ) = m + 8 if i = m m + i + 5 if m + i m, λ(v n v 0 ) = m +, λ(v 0 v) = m +, λ(vw) = m + 5. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case where m is even. (m + i)/ if i = 0,,..., m (m + i + )/ if i =,,..., m m if i = m + (m + i + )/ if i = m, m +,..., m (i m )/ if i = m +, m + 5,..., m (m + )/ if i = m +, λ(v i v i+ ) = λ(v) = (m + 8)/, λ(w) = (m + )/, m i + if i = 0,,..., m m + if i = m m i + 0 if i = m, m + 5m i + 0 if i = m +, m +,..., m m + if i = m, λ(v n v 0 ) = m + 8, λ(v 0 v) = m +, λ(vw) = m + 5. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Hence, the graph G is super edge-magic.

5 JI YEON PARK, JIN HYUK CHOI, AND JAE-HYEONG BAE Figure. An illustration for the labeling given in the proof of Proposition.. A complete graph is a simple graph in which every pair of vertices forms an edge. We denote by K n the complete graph with n vertices. Also, we denote by K C n the graph obtained from union of disjoint graphs K and C n. In [], W. D. Wallis proved that K C is not edge-magic, but K C is edgemagic. Then Wallis [] proposed the following problem: For which values of n, is K C n edge-magic? The following theorem shows that K C n is super edge-magic if n is even (n 0). Theorem.. Let G = K C n. The graph G is super edge-magic if n is even (n 0). Proof. Let v 0, v,..., v n, v 0 be a vertex sequence of C n and let u and w be the vertices of K. Let n = m for a nonnegative integer m. V (G) = {v 0, v,..., v n, u, w} and E(G) = {v i v i+ 0 i n } {v n v 0 } {uw}. We define a labeling λ : V E {,,..., ν + ϵ} as follows: If n =, then λ(v (C )) = (,, 5, ) and λ(v (K )) = {, }. Thus λ is a super edge-magic labeling of K C with the edge magic number. Now, consider n. Case m 0 (mod ) If m + i n, then m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m (m + 8)/ if i = m. (i + )/ if i (mod ) m + + i/ if i (mod ) (i + 5)/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i 5 (mod ) m + + i/ if i 0 (mod ),

6 ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Figure. An illustration for the labeling given in the proof of Theorem.. λ(u) = (m + )/, λ(w) = (m + )/, { m i + if 0 i m, λ(v i v i+ ) = m + if i = m. If m i n, then m + i if i (mod ) λ(v i v i+ ) = m + i if i (mod ) m i if i 0 (mod ), λ(v 0 v n ) = m +, λ(uw) = m +. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case m (mod ) m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m (m + 5)/ if i = m. If m + i n, then (i + )/ if i (mod ) m + + i/ if i (mod ) (i + 5)/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i 5 (mod ) m + + i/ if i 0 (mod ), λ(u) = (m + )/, λ(w) = (m + 5)/, { m i + if 0 i m λ(v i v i+ ) = m + if i = m.

7 JI YEON PARK, JIN HYUK CHOI, AND JAE-HYEONG BAE Figure. An illustration for the labeling given in the proof of Theorem.. If m i n, then m + i if i (mod ) λ(v i v i+ ) = m + i if i (mod ) m i if i 0 (mod ), λ(v 0 v n ) = m +, λ(uw) = m +. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case m (mod ) m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m (m + 5)/ if i = m. If m + i n, then (i + )/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i (mod ) m + + i/ if i (mod ) (i + 5)/ if i 5 (mod ) m + + i/ if i 0 (mod ) (m + )/ if i = m +, λ(u) = (m + )/, λ(w) = (m + 5)/, m i + if 0 i m λ(v i v i+ ) = m + if i = m m + i if m i m +.

8 ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Figure 5. An illustration for the labeling given in the proof of Theorem.. If m + i n, then m + i if i (mod ) λ(v i v i+ ) = m i if i (mod ) m + i if i 0 (mod ), λ(v 0 v n ) = m +, λ(uw) = m +. (See Figure 5 for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case m (mod ) m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m m/ + if i = m. If m + i n, then (i + )/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i (mod ) m + + i/ if i (mod ) (i + 5)/ if i 5 (mod ) m + + i/ if i 0 (mod ) m/ + if i = m +, λ(u) = (m + )/, λ(w) = (m + )/, m i + if 0 i m λ(v i v i+ ) = m + if i = m m + i if m i m +.

9 8 JI YEON PARK, JIN HYUK CHOI, AND JAE-HYEONG BAE Figure. An illustration for the labeling given in the proof of Theorem.. If m + i n, then m + i if i (mod ) λ(v i v i+ ) = m i if i (mod ) m + i if i 0 (mod ), λ(v 0 v n ) = m + λ(uw) = m +. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case m (mod ) m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m m/ + if i = m (m i + )/ if i = m +, m + m/ + if i = m + m i/ + 8 if i = m +, m +. If i = m + 5, m + i n, then (i + 5)/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i 5 (mod ) m + + i/ if i 0 (mod ), λ(u) = (m + )/, λ(w) = (m + )/,

10 ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS Figure. An illustration for the labeling given in the proof of Theorem.. λ(v i v i+ ) = m i + if 0 i m m + if i = m m + i if m i m + m if i = m + m + i if m + i m +. If m + i n, then m i if i (mod ) λ(v i v i+ ) = m + i if i (mod ) m + i if i 0 (mod ), λ(v 0 v n ) = m +, λ(uw) = m +. (See Figure for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Case m 5 (mod ) m + + i/ if i = 0,,..., m (i + )/ if i =,,..., m (m + 5)/ if i = m (m i + 0)/ if i = m +, m + (m + )/ if i = m + (m i + )/ if i = m +, m +. If i = m + 5, m + i n, then (i + 5)/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i (mod ) m + + i/ if i (mod ) (i + )/ if i 5 (mod ) m + + i/ if i 0 (mod ), λ(u) = (m + )/,

11 0 JI YEON PARK, JIN HYUK CHOI, AND JAE-HYEONG BAE Figure 8. An illustration for the labeling given in the proof of Theorem.. λ(w) = (m + 5)/, m i + if 0 i m m + if i = m λ(v i v i+ ) = m + i if m i m + m if i = m + m + i if m + i m +. If m + i n, then m i if i (mod ) λ(v i v i+ ) = m + i if i (mod ) m + i if i 0 (mod ), λ(v 0 v n ) = m +, λ(uw) = m +. (See Figure 8 for illustration.) It is easily seen that λ is a super edge-magic labeling of G with the edge magic number 5m +. Thus, the graph G is super edge-magic.. Closing remark We have shown that graph K C n is super edge-magic if n is even (n 0). It would be interesting to see if the graph is super edge-magic if n is odd. In fact, it seems to be very difficult to find families of graphs that are super edge-magic or are not. References [] H. Enomoto, A. S. Llado, T. Nakamigawa, and G. Ringel, Super edge-magic graphs, SUT J. Math. (8), no., [] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. (0), 5. [] G. Ringel and A. S. Llado, Another tree conjecture, Bull. Inst. Combin. Appl. 8 (), 8 85.

12 ON SUPER EDGE-MAGIC LABELING OF SOME GRAPHS [] W. D. Wallis, Magic Graphs, Birkhäuser Boston, Inc., Boston, MA, 00. [5] W. D. Wallis, E. T. Baskoro, M. Miller, and Slamin, Edge-magic total labelings, Australas. J. Combin. (000), 0. [] D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ,. Ji Yeon Park Department of Applied Mathematics Kyung Hee University Yongin -0, Korea address: Jin Hyuk Choi Department of Applied Mathematics Kyung Hee University Yongin -0, Korea address: Jae-Hyeong Bae Department of Applied Mathematics Kyung Hee University Yongin -0, Korea address: