On the super edge-magic deficiency of join product and chain graphs

A graph G of order |V (G)| = p and size |E(G)| = q is called super edge-magic if there exists a bijection f : V (G) ∪ E(G) → {1, 2, 3, · · · , p + q} such that f(x) + f(xy) + f(y) is a constant for every edge xy ∈ E(G) and f(V (G)) = {1, 2, 3, · · · , p}. Furthermore, the super edge-magic deficiency of a graph G, μs(G), is either the minimum nonnegative integer n such that G ∪ nK1 is super edge-magic or +∞ if there exists no such integer n. In this paper, we study the super edgemagic deficiency of join product of a graph which has certain properties with an isolated vertex and the super edge-magic deficiency of chain graphs.


Introduction
Let G be a finite and simple graph, where V (G) and E(G) are its vertex set and edge set, respectively.Let p = |V (G)| and q = |E(G)| be the number of the vertices and edges of G, respectively.Kotzig and Rosa [12] introduced the concepts of an edge-magic labeling and an edgemagic graph as follows: An edge-magic labeling of a graph G is a bijection f : V (G) ∪ E(G) → {1, 2, 3, • • • , p + q} such that f (x) + f (xy) + f (y) is a constant k, called the magic constant of www.ejgta.org On the super edge-magic deficiency of join product and chain graphs | A.A.G.Ngurah and R. Simanjuntak f , for every edge xy of G.A graph that admits an edge-magic labeling is called an edge-magic graph.Motivated by the concept of an edge-magic labeling, Enomoto et al. [6] introduced the concept of a super edge-magic labeling and a super edge-magic graph as follows: A super edgemagic labeling of a graph G is an edge-magic labeling f of G with the additional property that f (V (G)) = {1, 2, 3, • • • , p}.Thus, a super edge-magic graph is a graph that admits a super edgemagic labeling.The next lemma proved by Figueroa-Centeno et al. [7] provides necessary and sufficient conditions for a graph to be a super edge-magic graph.
Lemma 1.1.[7] A graph G is super edge-magic if and only if there exists a bijective function f : V (G) → {1, 2, • • • , p} such that the set S = {f (x) + f (y) : xy ∈ E(G)} consists of q consecutive integers.In this case, f can be extended to a super edge-magic labeling of G with the magic constant p + q + min(S).
The next lemma proved by Enomoto et al. [6] gives sufficient condition for non-existence of super edge-magic labeling of a graph.Lemma 1.2.[6] If G is a super edge-magic graph, then q ≤ 2p − 3.
In addition to these two lemmas, the notion of dual labeling will also appear frequently in the next sections.A dual labeling of a super edge-magic labeling f is defined as It has been proved in [4] that the dual of a super edge-magic labeling is also a super edge-magic labeling.Kotzig and Rosa [12] also proved that for every graph G there exists a nonnegative integer n such that G ∪ nK 1 is an edge-magic graph.This fact motivated them to introduced the concept of edge-magic deficiency of a graph.The edge-magic deficiency of a graph G, µ(G), is defined as the minimum nonnegative integer n such that G ∪ nK 1 is an edge-magic graph.Motivated by Kotzig and Rosa's concept of edge-magic deficiency, Figueroa-Centeno et al. [8] introduce the concept of super edge-magic deficiency of a graph.The super edge-magic deficiency of a graph G, µ s (G), is defined as either the minimum nonnegative integer n such that G ∪ nK 1 is a super edge-magic graph or +∞ if there exists no such n.
There have been a number of papers dealing with super edge-magic deficiency of graphs.In [1], Ahmad et al. studied the super edge-magic deficiency of some families related to ladder graphs and In [2], Ahmad et al. studied the super edge-magic deficiency of unicyclic graphs.In [11], Ichishima and Oshima investigated the super edge-magic deficiency of complete bipartite graphs and disjoint union of complete bipartite graphs.Other results can be found in [8,9] and the latest developments in these and other types of graph labelings can be found in the survey paper of graph labelings by Gallian [10].In this paper, we study the super edge-magic deficiency of join product graphs as well as the super edge-magic deficiency of some classes of chain graphs.

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On the super edge-magic deficiency of join product and chain graphs | A.A.G.Ngurah and R. Simanjuntak

Super edge-magic deficiency of join product graphs
Let G and H be vertex disjoint graphs.Join product of G and H, denoted by G + H, defined as a graph with Thus G + H is a graph of order p 1 + p 2 and size q 1 + q 2 + p 1 p 2 , where In this section, we study the super edge-magic deficiency of join product of a graph G which has certain properties with isolated vertices.Our first result gives necessary conditions for G + K 1 to have zero super edge-magic deficiency.
Lemma 2.1.Let G be a graph with no cycle and minimum degree one.If µ s (G + K 1 ) = 0 then G is a tree or a forest.
Proof.Let G be a graph of order p and size q.By Lemma 1.2, p+q ≤ 2(p+1)−3 or q ≤ p−1.
We also able to prove that the join product of some classes of trees and forests with an isolated vertex has zero super edge-magic deficiency as stated in Theorem 2.1.
e).For every n ≥ 1, µ s (DS n + K 1 ) = 0, where DS n is a double star.f).For every n ≥ 1 and m = 1, 2, µ s (G(n, m) + K 1 ) = 0, where G(n, m) is a graph obtained from K 1,n by attaching a path with m edges to a single leaf of K 1,n .
By a similar argument as in the proof of part a), the vertices of labels 1, 2 and 3 must form a triangle in H n or the vertex of label 1 is adjacent to the vertices of labels 2, 3 and 4. By these facts and since all triangles in H n have a common vertex z, then there are four following cases: Case 1.
. This vertex labeling can be extended to a super edge-magic labeling of H 2 with the magic constant 21.
) with (n + 2, 1, n + 3, n + 4) and label {x 1 , x 2 , . . ., x n } with {2, 3, . . ., n + 1}.This labeling can be extended to a super edge-magic labeling of H with magic constant 3n + 12.If x n+2 is removed, we get G(n, 1) + K 1 and the remaining labeling can be extended to a super edge-magic labeling of G(n, 1) The open problems relating to these results are as follows: Problem 1. Determine if the graphs [nP 2 ∪ P 3 ] + K 1 for n ≥ 7 and [nP 2 ∪ P 4 ] + K 1 for n ≥ 6 have zero super edge-magic deficiency.
As mentioned before, Figueroa-Centeno et al. [7] proved that µ s (F n ) = 0 if and only if 1 ≤ n ≤ 6.The natural question arise is what about the super edge-magic deficiency of join product of other trees of order at most six with an isolated vertex?In the next results, we study the super edge-magic deficiency of these graphs.Lemma 2.2.For any tree G of order p ≤ 6 excluding the tree in Figure 1 (a), µ s (G) = 0.
On the super edge-magic deficiency of join product and chain graphs | A.A.G.Ngurah and R. Simanjuntak Let H = G 1 + K 1 , where G 1 is the tree in Figure 1 It is not hard to prove that H is not super edge-magic.Furthermore, if we label z, x 1 , x 2 , x 3 , x 4 , x 5 , x 6 with 5, 7, 4, 1, 2, 8, 3, respectively, then this labeling can be extended to a super edge-magic labeling of H ∪ K 1 .So, µ s (H) = 1.The next result provides a sufficient condition of the join product of a tree of order p ≥ 7 with an isolated vertex to have nonzero super edge-magic deficiency.
Theorem 2.2.Let G be a tree of order p ≥ 7 and let Proof.Let µ s (H) = 0 with a super edge-magic labeling f .Since H is a graph of order p + 1 and size q = 2p − 1 = 2(p + 1) − 3, then S = {f (x) + f (y) : xy ∈ E(H)} = {3, 4, . . ., 2p + 1} and the vertices of labels 1, 2 and 3 must form a triangle or the vertex of label 1 is adjacent to the vertices of labels 2, 3 and 4, respectively.Also, the vertices of labels p + 1, p and p − 1 must form a triangle or the vertex of label p + 1 is adjacent to the vertices of labels p, p − 1 and p − 2, respectively.Since H is a graph of order p ≥ 8, the labels 1, 2, 3, 4, p + 1, p, p − 1 and p − 2 are all distinct.By combining these facts, we obtain either 2K 3 , K 3 ∪ K 1,3 or 2K 1,3 as a subgraph of H.However, 2K 3 cannot be a subgraph of H since every triangle in H share a common vertex.This completes the proof.
The converse of Theorem 2.2 is not true.To show this, let us consider the tree G 2 in Figure 1 (b).Define vertex and edge sets of G 2 + K 1 as follows: Then there exists a vertex labeling f such that 5f (z) + 3f (x 3 ) + f (x 2 ) + f (x 4 ) = 45.It is easy to check that any solutions of this equation do not lead to a super edge-magic labeling of G 2 +K 1 .So, µ s (G 2 + K 1 ) ≥ 1.If we label z, x 1 , x 2 , x 3 , x 4 , x 5 , y 1 and y 2 by 2, 3, 1, 6, 8, 4, 7 and 9, respectively, then this vertex labeling can be extended to a super edge-magic labeling of Next results provide the super edge-magic deficiency of join product of a tree with m ≥ 2 isolated vertices.Lemma 2.3.Let G a tree of order p ≥ 2 and m ≥ 2 be an integer.µ s (G + mK 1 ) = 0 if and only if G = P 2 .
On the super edge-magic deficiency of join product and chain graphs | A.A.G.Ngurah and R. Simanjuntak Lemma 2.3 show that µ s (G + mK 1 ) ≥ 1 for all the trees G = P 2 .Next lemma provides the lower bound of its super edge-magic deficiency.
Lemma 2.4.Let G be a tree of order p ≥ 3.For every positive integer m ≥ 2, Proof.This result is a corollary of the result of Ngurah and Simanjuntak [16] (see Lemma 2.2).

Super edge-magic deficiecy of chain graphs
Barrientos [3] defined a chain graph as a graph with blocks B 1 , B 2 , • • • , B k such that for every i, B i and B i+1 have a common vertex in such a way that the block-cut-vertex graph is a path.We denote the chain graph with Some authors have studied the super edge-magic deficiency of chain graphs.In 2003, Lee and Wang [13] proved that some classes of chain graphs whose blocks are complete graphs are super edge-magic.In other words, they showed that some classes of chain graphs whose blocks are complete graphs have zero super edge-magic deficiency.In [15], Ngurah et al. studied the super edge-magic deficiency of kK 3paths and kK 4 -paths.
Let L n = P n × P 2 be a ladder.Let TL n be the graph obtained from the ladder L n by adding a single diagonal in each rectangle of L n and let DL m be the graph obtained from the ladder L m by adding two diagonals in each rectangle of L m .It is clear that TL n is graph of order 2n and size 4n − 3 meanwhile DL m has 2m vertices and 5m − 4 edges.In this section, we study the super edge-magic deficiency of chain graphs where its blocks are combination of TL n and DL m .