More classes of super cycle-antimagic graphs - The Australasian ...

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 67(1) (2017), Pages 46–64

More classes of super cycle-antimagic graphs P. Jeyanthi∗ Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur, Tamilnadu India [email protected]

N.T. Muthuraja Department of Mathematics, Cape Institute of Technology Levengipuram, Tamilnadu India [email protected]

ˇova ´-Fen ˇovc ˇ´ıkova ´ A. Semanic Department of Applied Mathematics and Informatics Technical University, Letn´ a 9, Koˇsice Slovakia [email protected]

S. J. Dharshikha Deloitte Consulting US India Hyderabad India [email protected]

Abstract A simple graph G = (V, E) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to a given graph H. Then the graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for all subgraphs ∗

Corresponding author.

P. JAYANTHI ET AL. / AUSTRALAS. J. COMBIN. 67 (1) (2017), 46–64

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H of G isomorphic to H, the H -weights, wtf (H ) = v∈V (H ) f (v) + e∈E(H ) f (e), form an arithmetic progression a, a + d, a + 2d, . . . , a + (t − 1)d where a is the first term, d is the common difference and t is the number of subgraphs of G isomorphic to H. Such a labeling is called super if f (V ) = {1, 2, . . . , |V |}. This paper deals with some results on anti-balanced sets and we show the existence of super (a, d)-cycle-antimagic labelings of fans and some square graphs.

1

Introduction

Let G = (V, E) be a finite simple graph. A family of subgraphs H1 , H2 , . . . , Hn of G is called an edge-covering of G if each edge of E belongs to at least one of the subgraphs Hi , i = 1, 2, . . . , n. Then the graph G admitting an H-covering is (a, d)H-antimagic if there exists a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for all subgraphs H of G isomorphic to H, the H -weights, f (v) + f (e), wtf (H ) = v∈V (H )

e∈E(H )

form an arithmetic progression a, a + d, . . . , a + (t − 1)d where a > 0 is the first term, d ≥ 0 is the common difference and t is the number of subgraphs of G isomorphic to H. Such a labeling is called super if f (V ) = {1, 2, . . . , |V |}. For d = 0 it is called H-magic and H-supermagic, respectively. The concept of H-magic graphs was introduced by Gutiérrez and Lladó [7] as an extension of the edge-magic and super edge-magic graphs. They proved that some classes of connected graphs such as the stars K1,n and the complete bipartite graphs Kn,m are K1,h -supermagic for some h. They also proved that the path Pn and the cycle Cn are Ph -supermagic for some h. Lladó and Moragas [13] proved that wheels, windmills, books and prisms are Ch -magic for some h. Maryati et al. [17] and Salman et al. [20] proved that certain families of trees are path-supermagic. Jeyanthi and Selvagopal [10] proved that one point union of n copies of a 2-connected graph, linear garland of a 2-connected graph are H-supermagic. Interestingly, windmill is a particular case of one point union whereas ladder and triangular ladder are the particular cases of linear garland. Ngurah, Salman and Susilowati [19] proved that chains, wheels, triangles, ladders and grids are cycle-supermagic. Maryati, Salman and Baskoro [16] investigated the G-supermagicness of a disjoint union of c copies of a graph G and showed that the disjoint union of any paths is cPh -supermagic for some c and h. Muthuraja, Selvagopal and Jeyanthi [18] showed that the square graphs of bistar, path and cycle are cycle-supermagic. They also proved that the middle graph of Cn is also C3 -supermagic. Jeyanthi and Muthuraja [12] proved that the graph Pm,n for m, n ≥ 2 is C2m -supermagic and the splitting graph of Cn is C4 -supermagic for n = 4.


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The (a, d)-H-antimagic labeling was introduced by Inayah, Salman and Simanjuntak [8]. In [9] they investigated the super (a, d)-H-antimagic labelings for some shackles of a connected graph H. In [21] it is proved that wheels Wn , n ≥ 3, are super (a, d)-Ck -antimagic for every k = 3, 4, . . . , n − 1, n + 1 and d = 0, 1, 2. The (super) (a, d)-H-antimagic labeling is related to a super d-antimagic labeling of type (1, 1, 0) of a plane graph that is the generalization of a face-magic labeling introduced by Lih [14]. Further information on super d-antimagic labelings can be found in [2, 5]. For H ∼ = K2 , (super) (a, d)-H-antimagic labelings are also called (super) (a, d)edge-antimagic total labelings and have been introduced in [22]. More results on (a, d)-edge-antimagic total labelings, can be found in [4, 15]. The vertex version of these labelings for generalized pyramid graphs is given in [1]. The existence of super (a, d)-H-antimagic labelings for disconnected graphs is studied in [6] where it is proved that if a graph G admits a (super) (a, d)-H-antimagic labeling, where d = |E(H)|−|V (H)|, then the disjoint union of m copies of the graph G, denoted by mG, admits a (super) (b, d)-H-antimagic labeling as well. In [3] it is shown that the disjoint union of multiple copies of a (super) (a, 1)-tree-antimagic graph is also a (super) (b, 1)-tree-antimagic. A natural question is whether the similar result holds also for another differences and another H-antimagic graphs. A fan Fn , n ≥ 2, is a graph obtained by joining all the vertices of the path Pn on n vertices to a further vertex, called the centre. The vertices on the path we will call the path vertices. The edges adjacent to the central vertex are called the spokes and the remaining edges are called the path edges. The Fn contains n + 1 vertices and 2n − 1 edges. For a simple connected graph G, the square of the graph G, denoted by G2 , is defined as the graph with the same vertex set as G and two vertices are adjacent in G2 if they are at a distance 1 or 2 apart in G. In this paper we investigate the existence of super (a, d)-cycle-antimagic labelings of fans and some square graphs.

2

Known results on (k, δ)-anti-balanced sets

We use the following notation. For two integers a, b, a < b, let [a, b] denote the set of all integers from a to b. For any subset S of the set of integers Z we write, S = x∈S x and for an integer k, let k + S = {k + x : x ∈ S}. Thus k + [a, b] isthe set {x ∈ Z : k + a ≤ x ≤ k + b}. It can be easily verified that (k + S) = k|S| + S. If P = {X1 , X2 , . . . , Xn } is a partition of a set X of integers with the same cardinality then we say P is an n-equipartition of X. Also we denote the set of subsets sums of the parts of P by P = { X1 , X2 , . . . , Xn }. A multiset is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multisets elements. If X and Y are two multisets then their


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union is also a multiset represented by X Y . If an element a appears m times in X and n times in Y , then a appears m + n times in X Y . If X is a multiset of integers and k is an integer then k ⊕ X = {k + x : x ∈ X}. Let k ∈ N and let X be a multiset containing positive integers. Then X is said to be (k, δ)-anti-balanced if there exist k subsets of X, say X1 , X2 , . . . , Xk such that k , X = X and for i ∈ [1, k − 1], X − Xi = δ for every i ∈ [1, k], |Xi | = |X| i i+1 k i=1

is satisfied.

We use the following results to prove our main results. Lemma 2.1. h [7] Let h and k be two positive integers and let n =hk. For each integer 0 ≤ t ≤ 2 there is a k-equipartition P of [1, n] such that P is an arithmetic progression of difference d = h − 2t. Lemma 2.2. [11] If h is even, then there exists a k-equipartition P = {X1 , X2 , . . . , Xr = h(hk+1) for 1 ≤ r ≤ k. Thus, the subset sums Xk } of X = [1, hk] such that 2 h(hk+1) . are equal and is equal to 2 Lemma 2.3. [11] Let h and k be two positive integers such that h is even and k ≥ 3 is odd. Then there exists a k-equipartition P = {X1 , X2 , . . . , Xk } of X = [1, hk] such + r for 1 ≤ r ≤ k. Thus, P is a set of consecutive that Xr = (h−1)(hk+k+1) 2 (h−1)(hk+k+1) integers given by P= + [1, k]. 2 The above lemma was proved using permutations on [1, k] in [11]. We can deduce the result for h = 2 which is that {Y1 , Y2, . . . , Yk } is a k-equipartition of [1, 2k], where k−2i+1 , k + 2i for 1 ≤ r ≤ k−1 , 2 2 Yi = 3k−2i+1 k+1 , 2i for 2 ≤ r ≤ k 2 and

Yi =

3k+1 2

+ i for 1 ≤ i ≤ k.

Lemma 2.4. [11] Let h and k be two even positive integers and h ≥ 4. If X = [1, hk + 1] − { k2 + 1}, there exists a k-equipartition P = {X1 , X2 , . . . , Xk } of X such 2 + r for 1 ≤ r ≤ k. Thus P is a set of consecutive integers that Xr = h k+3h−k−2 2 h2 k+3h−k−2 + [1, k]. 2 In the proof of this lemma, it is shown that {Z1 , Z2 , . . . , Zk } is a k-equipartition of [1, 2k + 1] − { k2 + 1}, where k + 1 − i, k + 1 + 2i for 1 ≤ r ≤ k2 , 2 Zi = 3k + 2 − i, 2i for k2 + 1 ≤ r ≤ k 2 and

Zi =

3k 2

+ 2 + i for 1 ≤ i ≤ k.


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Lemma 2.5. [9] Let k be an integer such that k ≥ 2. If [1, k + 1] [2, k] for k odd, X= k k [1, 2 ] [ 2 + 2, k + 1] [2, k + 1] for k even, then X is (k, 1)-anti-balanced. Lemma 2.6. [9] Let k be an integer such that k ≥ 2. If X = [1, k] [2, k + 1] then X is (k, 2)-anti-balanced.

3

New results on (k, δ)-anti-balanced sets

We prove the following lemmas which are useful to prove our main results. Lemma 3.1. Let n, r be positive integers, n ≥ 2, 2 ≤ r ≤ n − 1 and let m = m min{r, n − r + 1}. If X = [j, n − j + 1] [1, n], then X is (n − r + 1, r)-antij=2

balanced. Proof. For 1 ≤ i ≤ n − r + 1, define Xi = {i, i + 1, . . . , i + r − 1}. It can be easily n−r+1 Xi = X. Also Xi = (r−1)r + ri and hence X is verified that |Xi | = r and 2 (n − r + 1, r)-anti-balanced.

i=1

Illustration 3.1. For n = 9 and r = 4, let X =

4

[j, 9−j +1] [1, 9] = [2, 8] [3, 7]

j=2

[4, 6] [1, 9]. We have (6, 4)-anti-balanced subsets X1 = {1, 2, 3, 4}, X2 = {2, 3, 4, 5}, X3 = {3, 4, 5, 6}, X4 = {4, 5, 6, 7}, X5 = {5, 6, 7, 8} and X6 = {6, 7, 8, 9}. Then the subset sums are 10, 14, 18, 22, 26, 30.

Lemma 3.2. Let n, r be positive integers, n ≥ 2, 2 ≤ r ≤ n+1 . If X = [r, n − r + 2 1] [1, n], then X is (n − r + 1, 2)-anti-balanced, assuming that [l, k] = ∅ if l > k. Proof. For 1 ≤ i ≤ n − r + 1, define Xi = {i, i + r − 1}. It is easy to verify that n−r+1 |Xi | = 2 and Xi = X. Also Xi = r − 1 + 2i and hence X is (n − r + 1, 2)anti-balanced.

i=1

Illustration 3.2. For n = 10, r = 6 and X = [6, 5] [1, 10] = [1, 10], we have (5, 2)-anti-balanced subsets X1 = {1, 6}, X2 = {2, 7}, X3 = {3, 8}, X4 = {4, 9} and X5 = {5, 10}. For n = 10, r = 5 and X = [5, 6] [1, 10], we have (6, 2)-anti-balanced subsets X1 = {1, 5}, X2 = {2, 6}, X3 = {3, 7}, X4 = {4, 8}, X5 = {5, 9} and X6 = {6, 10}. Lemma

Let n ≥ 2, be a positive integer and let X = [1, n]. If r divides n, then 3.3. X is nr , r 2 -anti-balanced.


Proof. For 1 ≤ i ≤

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n , r

define Xi = {(i − 1)r + 1, (i − 1)r + 2, . . . ir}. Obviously

|Xi | = r and ∪ Xi = X. Also Xi = r(r+1) + (i − 1)r 2 and hence X is nr , r 2 -anti2 i=1 balanced. n r

Illustration 3.3. Let n = 14, r = 2 and X = [1, 14]. We have (7, 4)-anti-balanced subsets X1 = {1, 2}, X2 = {3, 4}, X3 = {5, 6}, X4 = {7, 8}, X5 = {9, 10}, X6 = {11, 12} and X7 = {13, 14}. Lemma 3.4. Let n, r, r < n, be two relatively prime integers. Then the multiset r X = [1, n] is (n, 1)-anti-balanced. 1

Proof. Since gcd(n, r) = 1, the linear congruence rx ≡ 1 (mod n) has solutions. Let k be the solution such that k