Further Results on P h -supermagic Trees

—Let G be a simple, finite, and undirected graph. An H -supermagic labeling is a bijective map f : V ( G ) ∪ E ( G ) → { 1 , 2 , · · · , | V ( G ) | + | E ( G ) |} in which f ( V ) = { 1 , 2 , · · · , | V ( G ) |} and there exists an integer m such that w ( H ′ ) = (cid:80) v ∈ V ( H ′ ) f ( v ) + (cid:80) e ∈ E ( H ′ ) f ( e ) = m , for every subgraph H ′ ∼ = H in G . In this paper, we determine some classes of trees which have P h -supermagic labeling.

Further Results on P h -supermagic Trees Tita Khalis Maryati, Otong Suhyanto, and Fawwaz Fakhrurrozi Hadiputra Abstract-Let G be a simple, finite, and undirected graph.An H-supermagic labeling is a bijective map f : |} and there exists an integer m such that w(H ′ ) = v∈V (H ′ ) f (v) + e∈E(H ′ ) f (e) = m, for every subgraph H ′ ∼ = H in G.In this paper, we determine some classes of trees which have P h -supermagic labeling.

I. INTRODUCTION
L ET G be a simple, finite, and undirected graph.Let two isomorphic graphs G and G ′ are denoted by G ∼ = G ′ .A graph G is called a tree if it does not contain any cycles.An amalgamation of a collection of graphs {G i } is obtained by picking a vertex to be a terminal in each of G i , and identifying the graphs by their terminals [1].We use the notation Amal{(G i ), c} for an amalgamation which is obtained from identifying all c from each G i .For convenience, let Let H be a subgraph of G.If the graph G has a property that each edges of G belongs to at least one subgraph isomorphic to H in G, we say G admits H-covering.A bijection f : then f may also be called H-supermagic.The problem is to determine whether a certain graph admits H-magic or H-supermagic labeling.Some known results are found for the subgraph H is isomorphic to either a star K 1,n , a cycle C n , and a path P n .Gutiérrez and Lladó [2] Roswitha and Baskoro [3] have determined some double stars, caterpillars, fire crackers, and banana trees to be K 1,h -magic for some h.Moreover, some known graphs which are C h -magic (or supermagic) for some h are found which include wheels, windmills [4], fans, books, ladders [5], and jahangirs [6].
Furthermore, some results of P h -magic (or supermagic) graphs are also determined.Gutiérrez and Lladó [2] found that paths are P h -supermagic as follows.
Theorem 1: Let n ≥ 3 be an integer.The path Next, let h ≥ 3 be an integer.We define grass graph to be the class graph of trees that admits P h -covering such that all T. K. Maryati and O. Suhyanto are with Department of Mathematics Education, UIN Syarif Hidayatullah Jakarta, Indonesia e-mail: tita.khalis@uinjkt.ac.id, otong.suhyanto@uinjkt.ac.id.F. F. Hadiputra is with the Master Program of Mathematics, Institut Teknologi Bandung, Indonesia e-mail: fawwazfh@alumni.ui.ac.id.
subgraphs P h in the graph contain identical vertex, denoted by Rb(h).A center of grass graph is the identical vertex of every subgraph P h .We may write an equivalent theorem by [7] as follows.
Theorem 2: Let h ≥ 2 be an integer.Any graph belongs to Rb(h) is P h -supermagic.
The definition of Rb(h) may be used to find a radius of a tree graph.A radius of a tree graph r(G) may be defined as r(G) = m − 1 where m is the least number h such that the graph admits P h -covering.
Known results are discussing not only sufficient conditions of P h -magic graphs but also necessary conditions of P h -magic graphs.One of the results is determined by Maryati et al. [8] stating that P h -magic graphs cannot contain a subgraph H n constructed as follows.Let n ≥ 1 be an integer.Obtain two disjoint odd paths P 2n+1 and add one more edge such that the center of those two graphs are adjacent.
Theorem 3: Let n be a positive integer and h ∈ Some other P h -magic (or supermagic) graphs are shackles and amalgamations [1], disjoint union of graphs and amalgamations [9], and cycles with some pendants [8].Variants for this problem can be seen in [10], [11] and for more information of H-magic (or supermagic) labeling, please consult to [12].In this paper, we would like to investigate more about P hsupermagic tree graphs.

Denote e c
v to be an edge which belongs to a path from v to c and incident to v. We start this section by introducing an useful lemma.
Lemma 1: Let h ≥ 2 be an integer, and G be a P h -magic tree with a magic labeling f where t = |V (G)|.If there exists a subgraph H which belongs to Rb(h) with c as a center, such that every pair v i ∈ H and its incident edge e c vi satisfy or equivalent of total vertices in H ′ without its center.Let f ′ be a labeling of G ′ .Then, for every v ∈ V (G) and e ∈ E(G), label as follows Take all the unused labels {t+1, t+2, ..., t+2n} and create a partition into 2-sets, sets consists of two elements, such that the sum of the elements of each 2-set is 2t + 2n + 1.Then, use all these 2-sets to label all {v i , e c vi } in any order so that .
This lemma enable us to identify the center of any Rb(h) to a terminal vertex of P h -magic graph in order to produce other P h -magic (or supermagic) graph.The terminal vertex chosen for this study is mostly a pendant.
Theorem 4: Let h ≥ 3 be an integer and let H belongs to Rb(h).The graph G ∼ = Amal{(H, P h+1 , P h+1 ), c} with c is a center of H and a pendant of each P h+1 is P h -supermagic. Proof: Compile the unused labels {3} ∪ {7, ..., t} ∪ {t + 1, ..., 2t − 6} ∪ {2t − 2} and create a partition of 2-sets such that the sum of the elements of each 2-sets is 2t + 1.Then, use all these 2-sets to label all unlabeled pairs {v i , e c vi }, {u i , e c ui } and {q, e c q } for q ∈ H in any order such that , and f (q) < f (e c q ).By Lemma 1, G is P h -supermagic.
An example of a tree for Theorem 4 can be seen in Figure 1.Theorem 5: Let h ≥ 3 be an integer and let H belongs to Rb(h).The graph G ∼ = Amal{(H, P h+1 , P h+1 , P h+1 ), c} with c is a center of H and a pendant of each P h+1 is P hsupermagic. Proof: Compile the unused labels and create a partition of 2sets such that the sum of the elements of each 2-sets is 2t + 1.Then, use all these 2-sets to label all unlabeled pairs {u i , e c ui }, {v i , e c vi }, {x i , e c xi } and {q, e c q } for q ∈ H in any order such that ) and f (q) < f (e c q ).By Lemma 1, G is P h -supermagic.Theorem 6: Let h ≥ 3 be an integer and let H belongs to Rb(h).The graph G ∼ = Amal{(H, P h+1 , P h+2 ), c} with c is a center of H and a pendant both of P h+1 and P h+2 is P h -supermagic. Proof: Compile the unused labels and create a partition of 2sets such that the sum of the elements of each 2-sets is 2t + 1.Then, use all these 2-sets to label all unlabeled pairs {v i , e c vi }, {u i , e c ui } and {q, e c q } for q ∈ H in any order such that , and f (q) < f (e c q ).By Lemma 1, G is P h -supermagic.
Before we continue to the next theorem, we need define to define P + n .Let y be one of a vertices in a path P h which is adjacent to a pendant.For n ≥ 4, a graph P + n is obtained from a path P n which the vertex y is attached with one more pendant.A pendant z of P + n is called a furthest pendant if it is not adjacent to y.
Theorem 7: Let h ≥ 3 be an integer and let H of order at least two belongs to Rb(h).The graph G ∼ = Amal{(H, P + n , P n ), c} with c is a center of H, and a (furthest) pendant of both P h and P + h is P h -supermagic.Proof: Since the order of H is at least two, there exists a subgraph Compile the unused labels and create a partition of 2-sets such that the sum of the elements of each 2-sets is 2t + 1.Then, use all these 2-sets to label all unlabeled pairs {v i , e c vi }, {u i , e c ui }, and {q, e c q } for q ∈ H in any order such that , and f (q) < f (e c q ).By Lemma 1, G is P h -supermagic.
An example of a tree in Theorem 7 is illustrated in Figure 2. Theorem 8: Let h ≥ 2 be an integer and let H, H ′ belongs to Rb(h).If G ′ ∼ = Amal{(H, P h ), c}, where c is a center of H and a pendant of P h , then the amalgamation G ∼ = Amal{(G ′ , H ′ ), c ′ } where c ′ is a center of H ′ and the other pendant of P h is P h -supermagic.
Proof: First, we need to prove G ′ ∼ = Amal{(H, P h ), c} is P h -supermagic with a magic labeling satisfying the condition of the lemma.Denote Label the vertices and edges as follows Then, take {1, 2, 3, ..., t − h} ∪ {t + h, t + 2, ..., 2t − 1}, and create a partition of 2-sets such that the sum of the elements of each 2-set is 2t.Use all these 2-sets to label all {v i , e c vi } in any order so that where f (v i ) < f (e c vi ).By evaluating, for every P h we got w(P h ) = (h − 1)(2t) + f (c).
We have shown that G ′ ∼ = Amal{(H, P h ), c} is P hsupermagic with a magic labeling f .It can be seen that there exists a subgraph H * of G ′ which belongs to Rb(h) with u h as a center.This subgraph and the magic labeling f are satisfying Lemma 1, hence by applying the lemma, we have Amal{ Furthermore, the next result is applicable for h = 3. Theorem 9: Let H 1 , H 2 be graphs which belongs to Rb(3).Then, Amal{(H 1 , H 2 ), p} where p is a pendant of both H and H ′ is P 3 -supermagic. Proof: Create a partition for A and B into 2-sets such that the sum of the elements of each 2-set is 2(t 1 + t 2 − 1) for A and 2(t 1 + t 2 ) − 1 for B. Construct a f labeling as follows Use all 2-sets from A to label all {v, e c1 v } for v ∈ H 1 in any order so that . Again, use all 2-sets from B to label all {u, e c2 u } for u ∈ H 2 in any order so that f (u) + f (e c2 u ) = 2(t 1 + t 2 ) − 1 where f (u) < f (e c2 u ).Therefore, every vertices have smaller labels from every edges.Furthermore, for every P 3 we got f (P 3 ) = 4(t 1 + t 2 ) + 6.
Create a partition for each A and B into 2-sets such that the sum of the elements of each 2-set is 2(t 1 + t 2 − 1) for A and 2(n 1 + n 2 ) − 3 for B. Construct a f labeling as follows f (e c1 p ) = t 1 + 2t 2 , f (e c2 p ) = t 1 + 2t 2 − 1 Choose v 1 ∈ H 1 other than c 1 or p. Continue labels as follows f (e c1 v1 ) = t 1 + t 2 .Use all 2-sets from A to label all {v, e c1 v } for v ∈ H 1 , v ̸ = v 1 in any order so that f (v) + f (e c1 v ) = 2(t 1 + t 2 − 1) with f (v) < f (e c1 v ).Again, use all 2-sets from B to label all {u, e c2 u } for u ∈ H 2 in any order so that f (u) + f (e c2 u ) = 2(t 1 + t 2 ) − 3 with f (u) < f (e c2 u ).Therefore, every vertices have smaller labels from every edges.Furthermore, for every P 3 we got f (P 3 ) = 5(t 1 + t 2 ) − 7.
Hence, the theorem holds.