SUPER EDGE ANTI-MAGIC VERTEX LABELING AND SUPER EDGE ANTI-MAGIC TOTAL LABELING OF A CYCLE WITH ZIG-ZAG CHORDS

In this paper, we show that super edge anti-magic vertex labeling and super edge anti-magic total labeling of cycles by adding (n-3) zig-zag chords. AMS Subject Classification: 05C78


Introduction
All graphs are finite, simple and undirected.The graph G has vertex-set V(G) and edge-set E(G).Unless otherwise noted, |V (G)| = p and |E(G)| = q.
A labeling of a graph is any map that carries some set of graph elements to numbers (usually to the positive or non-negative integers).Magic labelings are one-to-one maps onto the appropriate set of consecutive integers starting from 1, with some kind of "constant-sum" property.
Simanjuntak, Miller and Bertault [8] defined an (a, d)-edge-antimagic vertex ((a, d)-EAV) labeling for a graph G(V,E) as an injective mapping f from V onto the set {1, 2, • • • , n} with the property that the edge-weights {w(xy) : w(xy) = f (x) + f (y), xy ∈ E}, form an arithmetic sequence with the first term a and difference d, where a > 0 and d ≥ 0 are two fixed integers.An (a, d)-EAV labeling is called super (a, d)-edge antimagic vertex ((a, d)-SEAV) labeling if Acharya and Hegde [1] (see also [6]) introduced the concept of a strongly (a, d)-indexable labeling which is equivalent to (a, d)-EAV labeling.The relationship between the sequential graphs and the graphs having an (a, d)-EAV labeling is shown in [3].
An (a, d)-edge antimagic total ((a, d)-EAT) labeling is a bijection f from V ∪ E onto {1, 2, • • • , v + e} with the property that the sums of the label on the edges and the labels of their end points form an arithmetic sequence starting from a and having a common difference d.This labeling is a natural extension of the notion of edge magic labeling which was originally introduced by Kotzig and Rosa in [7], where edge-magic labeling is called magic valuation.Relationships between (a, d)-EAT labeling and other labelings, namely, (a, d)-EAV labeling are presented in [2].
An (a, d)-EAT labeling is called super (a, d)-edge antimagic total ((a, d)-SEAT) labeling if f (V ) = 1, 2, • • • , n.This labeling is a natural extension of the notion of a super edge-magic labeling defined by Enomoto et al. in [4].A graph that has an (a, d)-EAV ((a, d)-EAT or (a, d)-SEAT) labeling is called an (a, d)-EAV ((a, d)-EAT or (a, d)-SEAT) graph.J.A. MacDougall and W.D. Wallis conjecture in [9] that every cycle with a chord is strong edge magic labeling.Also M. Baca and M. Murugan conjecture in [10] that every cycle with a chord is super edge-antimagic labelling.
A cycle with (n-3) chords is a graph obtained from a cycle C n (n ≥ 4) by adding (n-3) zig-zag chords to the non-consecutive vertices of the cycle, that is between each pair of non adjacent vertices 2 for odd cycles.

Main Results
Theorem 1.A cycle C n for n ≥ 4 with (n-3) zigzag chords has a (3,1)super edge anti-magic vertex labeling.
Proof.Let C n be a cycle on n vertices.We denote the vertices of C n as the clockwise direction and denote the edges of C n with (n-3) zigzag chords as e 1 , e 2 , e 3 , • • • , e (2n−3) such that e i = v i v (i+1) for 1 ≤ i ≤ (n−1), e n = v n v 1 and for chords, when n is even when n is odd The labeling for the vertices of C n are given as follows.Define From the above definition it is observed that the vertices of C n are labeled from 1 to n and are distinct.Now the edge-weights can be labelled as, when n is even, the edge labeling of chords are when n is odd, the edge labeling of chords are, Hence the cycle C n with (n-3) zigzag chords has a (3,1)-super edge anti-magic vertex labeling.
Theorem 2. A cycle C n for n ≥ 4 with (n-3) zigzag chords has a (n+4,2)super edge anti-magic total labelling.Proof.Let C n be a cycle on n vertices.We denote the vertices of when n is odd The labeling for the vertices of C n are given as follows.Define From the above definition it is observed that the vertices of C n are labeled from 1 to n and are distinct.Now the edge-weights can be labelled as, when n is even, the edge labeling of chords are, when n is odd, the edge labeling of chords are, f (e 3n+2i−7 2 Hence the cycle C n with (n-3) zigzag chords has a (n+4,2)-super edge anti-magic total labeling.
the clockwise direction and denote the edges of C n with n-3 zigzag chords as e 1 , e 2 , e 3 , • • • , e 2n−3 such that e