Ratio Mathematica Volume 42, 2022 D4-Magic Graphs Anusha Chappokkil* Anil Kumar Vasu† Abstract Consider the set X = {1, 2, 3, 4} with 4 elements. A permutation of X is a function from X to itself that is both one one and on to. The permutations of X with the composition of functions as a binary op- eration is a nonabelian group, called the symmetric group S4. Now consider the collection of all permutations corresponding to the ways that two copies of a square with vertices 1, 2, 3 and 4 can be placed one covering the other with vertices on the top of vertices. This col- lection form a nonabelian subgroup of S4, called the dihedral group D4. In this paper, we introduce A-magic labelings of graphs, where A is a finite nonabelian group and investigate graphs that are D4-magic. This did not attract much attention in the literature. Keywords: A-magic labeling; Dihedral group D4 ; D4-magic. 2020 AMS subject classifications: 05C25, 05C78 .1 *Department of Mathematics, University of Calicut, Malappuram, Kerala, India 673 635.; canusha235@gmail.com. †Department of Mathematics, University of Calicut, Malappuram, Kerala, India 673 635.; anil@uoc.ac.in 1Received on March 22th, 2022. Accepted on June 12th, 2022. Published on June 30th 2022. doi: 10.23755/rm.v41i0.738. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 167 C. Anusha and V. Anil Kumar 1 Introduction A graph G is an ordered pair (V (G), E(G)), where V (G) is a finite nonempty set whose elements are called vertices and E(G) is a binary irreflexive and sym- metric relation on V (G) whose elements are called edges. For any abelian group A, written additively, any mapping ℓ : E(G) → A \ {0} is called a labeling. Given a labeling on the edge set E(G), one can introduce a vertex set labeling ℓ+ : V (G) → A as follows: ℓ+(v) = ∑ uv∈E(G) l(uv) A graph G is said to be A-magic if there is a labeling ℓ : E(G) → A\{0} such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant, that is, ℓ+(v) = a for some fixed a ∈ A. The original concept of A-magic graph was introduced by Sedláček[1]. According to him, a graph G is A-magic if there exists an edge labeling on G such that (i) distinct edges have distinct non-negative labels; and (ii) the sum of the labels of the edges incident to a particular vertex is same for all vertices. When A = Z, the Z-magic graphs are considered in Stanley[7]. Doob [5, 4] also considered A-magic graphs where A is an abelian group. Also he determined which wheels are Z-magic. Observe that several authors studied V4-magic graphs[8, 6]. It is natural to ask does there exist graphs which admits A-magic labeling, when A is nonabelian? In this paper, we address this question and investigate graphs that are D4-magic. 2 Main results Let G = (V (G), E(G)) be a finite (p, q) graph and let (A, ∗) be a finite non- abelain group with identity element 1. Let f : E(G) → Nq = {1, 2, . . . , q} and let g : E(G) → A\{1} be two edge labelings of G such that f is bijective. Define an edge labeling ℓ : E(G) → Nq × A \ {1} by l(e) := (f(e), g(e)), e ∈ E(G). Define a relation ≤ on the range of ℓ by: (f(e), g(e)) ≤ (f(e ′ ), g(e′)) if and only if f(e) ≤ f(e′). Then obviously the relation ≤ is a partial order on the range of ℓ. Let {(f(e1), g(e1)), (f(e2), g(e2)), . . . , (f(ek), g(ek))} be a chain in the range of ℓ. We define the product of elements of this chain as follows: k∏ i=1 (f(ei), g(ei)) := ((((g(e1) ∗ g(e2)) ∗ g(e3)) ∗ g(e4)) ∗ . . .) ∗ g(ek). 168 D4 magic graphs Let u ∈ V and let N∗(u) be the set of all edges incident with u. Note that the range of ℓ|N∗(u) is a chain, say (f(e1), g(e1)) ≤ (f(e2), g(e2)) ≤ · · · ≤ (f(en), g(en)). We define, ℓ∗(u) = n∏ i=1 (f(ei), g(ei)). (1) If ℓ∗(u) is a constant, say a for all u ∈ V (G), we say that the graph G is A- magic. The map ℓ∗ is called an A-magic labeling of G and the corresponding constant a is called the magic constant. For example, consider the cycle graph C4 = (uv, vw, wx, xu) and the permutation group D4. Note that the group D4 is a non abelian group of order 8 and its elements are given by ρ0 = ( 1 2 3 4 1 2 3 4 ) , µ1 = ( 1 2 3 4 4 3 2 1 ) , ρ1 = ( 1 2 3 4 2 3 4 1 ) , µ2 = ( 1 2 3 4 2 1 4 3 ) , ρ2 = ( 1 2 3 4 3 4 1 2 ) , δ1 = ( 1 2 3 4 1 4 3 2 ) , ρ3 = ( 1 2 3 4 4 1 2 3 ) , δ2 = ( 1 2 3 4 3 2 1 4 ) . Define f : E(G) → N4 = {1, 2, 3, 4} as f(uv) = 1, f(wx) = 2, f(vw) = 3, f(xu) = 4 and g : E(G) → D4 \ {ρ0} as g(uv) = g(wx) = ρ1, g(vw) = g(xu) = δ1. Thus ℓ∗(u) = (1, ρ1)(4, δ1) = ρ1δ1 = µ2, ℓ∗(v) = (1, ρ1)(3, δ1) = ρ1δ1 = µ2. Similarly, ℓ∗(w) = µ2 and ℓ∗(x) = µ2. Thus C4 is D4-magic with magic constant µ2. Figure 1: D4-magic labeling of C4. In this paper, we will consider the symmetric group D4 and investigate graphs that are D4-magic. 169 C. Anusha and V. Anil Kumar Theorem 2.1. Let A be a non abelian group having an element of order 2 and let G be a graph. If either the degree of the vertices of G are all even or odd. Then G is A-magic. Proof. Let G be a (p, q) graph and A be a nonabelian group having an element of order 2. Let a ∈ G is of order 2. Let g : E(G) → A \ {1} be the constant map g(e) = a, ∀e ∈ E(G) and let f be any bijection from E(G) → Nq. First assume that all the vertices of G are of even degree then l∗(u) = 1, ∀u ∈ V (G). Similarly, if all the vertices of G are of odd degree then l∗(u) = a, ∀u ∈ V (G). Hence the proof. Corolary 2.1. All Eulerian graphs are D4-magic. Theorem 2.2. Any regular graph is D4-magic. Proof. Let G = (V (G), E(G)) be a regular graph with |E(G)| = q. Let f : E(G) → Nq be any bijection and g be any constant map from E(G) → D4\{ρ0}. Obviously, f and g will determine a D4-magic labeling of G. This completes the proof of the theorem. Corolary 2.2. For any n ≥ 3, the cycle graph Cn is D4-magic. Corolary 2.3. For any n ≥ 2, the complete graph Kn is D4-magic. Corolary 2.4. The Peterson graph is D4-magic. Theorem 2.3. The star graph K1,n, n ≥ 2 is D4-magic iff n is odd. Proof. Let G = K1,n. Suppose that n is odd. Let f : E(G) → Nn+1 be a bijection. Define g : E(G) → D4 \ {ρ0} by g(e) = µ1. Then clearly it is D4- magic with magic constant µ1. Conversely, suppose K1,n is D4-magic with magic constant, say ‘a’. So every pendent edge of K1,n should be mapped to a under g. Let u be the vertex of K1,n with degree n. Then ℓ∗(u) = aa · · · a︸ ︷︷ ︸ n times = a. This implies that an−1 = ρ0. If n is odd, the equation an−1 = ρ0 has five non trivial solutions in D4 viz. µ1, µ2, δ1, δ2 and ρ2. On the other hand, if n is even there are no element in D4 such that an−1 = ρ0. This completes the proof. A bistar graph Bn is the graph obtained by connecting the apex vertices of two copies of star K1,n by a bridge. Theorem 2.4. The bistar graph Bn, n > 1 is D4-magic when n ̸≡ 1(mod 4). 170 D4 magic graphs Proof. First, observe that there are 2n pendant edges and one bridge in Bn. Here we consider the following cases: Case (i): n is even (n ≡ 2(mod 4) or n ≡ 0(mod 4)). If n is even, define g : E(Bn) → D4 \ {ρ0} by g(e) = µ1, ∀e ∈ E(Bn). Let f be any bijective map from E(Bn) → N2n+1. Then obviously, Bn is D4-magic with magic constant µ1. Case (ii): n ≡ 3(mod 4). In this case we define g : E(Bn) → D4 \ {ρ0} by g(e) = { ρ1, if e is a pendant edge, ρ2, if e is the bridge. Let f be any bijective map from E(Bn) to N2n+1. Then obviously Bn is D4-magic with the magic constant ρ1. Case (iii): n ≡ 1(mod 4). Suppose that n ≡ 1(mod 4). Let k1 and k2 be the apex vertices of the bistar graph. Assume that Bn is D4-magic with magic constant µ1. Therefore, g(e) = µ1 for all pendant edges e. Assume that g(k1k2) = a, where a ∈ D4 \ {ρ0}. Without loss of generality assume that f(k1k2) > f(b), ∀b ∈ E(G), where b denotes the pendant edge with one end point k1. Then ℓ∗(k1) = µ1µ1 . . . µ1︸ ︷︷ ︸ (n times) a = µ1. The above equation tells us that a = ρ0, which is a contradiction. This contradiction shows that Bn is not D4-magic with magic constant µ1. In a similar manner, we can prove that Bn is not D4-magic with magic constants µ2, ρ1, ρ2, ρ3, δ1 or δ2. Thus the bistar graph Bn is not D4-magic when n ≡ 1(mod 4). This completes the proof of the theorem. Theorem 2.5. The complete bipartite graph Km,n is D4-magic, m, n > 1. Proof. Let G = Km,n. Suppose U = {u1, u2, . . . , un}, and V = {v1, v2, . . . , vm}. be the two partite sets of Km,n. If m and n are both even or odd then the theorem is obvious by taking any constant map g : E(G) → {ρ2, µ1, µ2, δ1, δ2}. Case (i): n ≡ 0(mod 2) and m ≡ 1(mod 4). Let U = {u1, u2, . . . , u2l} and V = {v1, v2, . . . , v4r+1} where n = 2l, m = 4r + 1, and l, r ∈ N. For 1 ≤ i ≤ n and 1 ≤ j ≤ m define g(uiv5k+1) = µ1, where k < m, k = 0, 1, 2, 3, . . . g(uiv5k+2) = µ2, k < m, k = 0, 1, 2, 3, . . . g(uivj) = ρ2, j ̸= 5k + 1, 5k + 2 where k = 0, 1, 2, . . . f(uivj) = (i − 1)m + j, 1 ≤ i ≤ n, 1 ≤ j ≤ m. 171 C. Anusha and V. Anil Kumar The maps f and g will determine a D4-magic labeling for Km,n with magic constant ρ0. Case (ii): n ≡ 0(mod 2) and m ≡ 3(mod 4). Define g as follows: g(uivj) = { ρ1, if i is odd 1 ≤ j < m, ρ3, if i is even 1 ≤ j < m, and g(uivm) = ρ2, ∀i, 1 ≤ i ≤ n, and let f be any bijection from E(G) to {1, 2, . . . , mn}. Then clearly f and g will determine a D4-magic labeling of Km,n with magic constant ρ0. This completes the proof of the theorem. 3 Cycle Generated Graphs In this section, we consider certain graphs which are constructed from cycles. A wheel Wn of order n + 1, sometimes simply called an n wheel is a graph that contains a cycle of order n and for which every graph vertex in the cycle is connected to one other graph vertex (which is known as the hub). The edges of a wheel which include the hub are called spokes. The wheel Wn can be defined as the graph join K1 + Cn, where K1 is the singleton graph and Cn is the cycle graph. Theorem 3.1. If n ≥ 3, the wheel Wn is D4-magic. Proof. Let the vertices of Cn be u1, u2, . . . , un such that uiui+1 ∈ E(Cn), i = 1, 2, . . . , n and un+1 = u1. Denote the vertex of K1 by k. Now we consider the following cases: Case (i): n is odd. If n is odd then every vertex of Wn is of odd degree. Thus we can take g : E(Wn) → D4\{ρ0} as any constant map from E(Wn) to {ρ2, µ1, µ2, δ1, δ2}. Since g is constant we can take f as any bijection from E(Wn) to N2n. Clearly this f and g will constitute a D4-magic labeling for Wn. Case (ii): n is even. Suppose n is even define f : E(Wn) → N2n as f(kui) = i, i = 1, 2, . . . , n, f(uiui+1) = n + i, 1 ≤ i ≤ n − 1, f(u1un) = 2n. 172 D4 magic graphs Now we can define g : E(Wn) → D4 \ {ρ0} by labeling each spokes by µ1 and all the outer edges by µ2 and ρ2 alternatively. Then Wn becomes D4-magic with magic constant ρ0. This completes the proof of the theorem. The helm Hn is a graph obtained from a wheel Wn by attaching a pendant edge at each vertex of the n cycle. Theorem 3.2. The Helm graph Hn is D4-magic. Proof. Let {k, ui, vi : i = 1, 2, . . . , n} be the vertex set of Hn, where k be the central vertex, u1, u2, . . . , un are the vertices of the cycle, v1, v2, . . . , vn are the pendant vertices adjacent to u1, u2, . . . , un. The edge set of Hn is E(Hn) = {uiui+1, kui, uivi : i = 1, 2, . . . , n, un+1 = u1}. Now consider the following two cases: Case(i): n is odd. Suppose that n is odd. Define f and g as follows: Let g : E(G) → D4\{ρ0} be defined as g(kui) = ρ2, 1 ≤ i ≤ n, g(ujuj+1) = ρ1, 1 ≤ j ≤ n − 1, g(u1un) = ρ1, g(ukvk) = ρ2, 1 ≤ k ≤ n. Now let f : E(G) → N2n+1 be any bijection. Then clearly f and g will give a D4-magic labeling of Hn, where n is odd. Case(ii): n is even. Let f be defined as above and define g : E(G) → D4 \ {ρ0} by g(uivi) = ρ2, 1 ≤ i ≤ n, g(v1vn) = ρ1 g(kuj) = { ρ2, if 1 ≤ j ≤ n − 2, ρ1, if j = n − 1, n. , g(ukuk+1) = { ρ1, if 1 ≤ k ≤ n − 2, ρ2, if k = n − 1. It follows that l∗(u) = ρ2, ∀u ∈ V (G). Hence Hn is D4-magic when n is even. This completes the proof of the theorem. The web graph W(2, n) is a graph obtained joining the pendant points of a helm to form a cycle and adding a single pendant edge to each vertex of this outer graph. Theorem 3.3. The web graph W(2, n), n ≥ 3 is D4-magic. 173 C. Anusha and V. Anil Kumar Proof. Let {k, ui, vi, wi : i = 1, 2, 3, . . . , n} be the vertex set of W(2, n), where k be the central vertex, u1, u2, u3, . . . , un are the vertices of inner cycle, v1, v2, v3, . . . , vn are the vertices of outer cycle and w1, w2, w3, . . . , wn are the pendant vertices adjacent to v1, v2, v3, . . . , vn of W(2, n). Let E(W(2, n)) = {uiui+1, vivi+1, uivi, viwi : i = 1, 2, . . . , n and un+1 = u1, vn+1 = v1}. We define a D4-magic labeling for W(2, n) with magic constant ρ2 as follows: Case (i): n is odd. Let f : E(G) → N3n+1 be any bijection. Define g : E(G) → D4 \ {ρ0} as g(kui) = ρ2 = g(uivi) = g(viwi), 1 ≤ i ≤ n, g(uiui+1) = ρ1 = g(vivi+1), 1 ≤ i ≤ n − 1, g(u1un) = ρ1 = g(v1vn). Case (ii): n is even. Let f : E(G) → N3n+1 be any bijection. Define g : E(G) → D4 \ {ρ0} as g(kui) = ρ2, 1 ≤ i < n − 1, g(kun) = g(kun−1) = ρ1, g(vivi+1) = ρ1 = g(uiui+1) = ρ1, 1 ≤ i ≤ n − 1, g(viwi) = ρ2 = g(uivi), 1 ≤ i ≤ n, g(v1vn) = ρ1, g(u1un) = ρ2. This completes the proof of the theorem. A shell graph Sn,n−3 of width n is a graph obtained by taking n−3 concurrent chords in a cycle Cn of n vertices. The vertex at which all chords are concurrent is called is called the apex. The two vertices adjacent to the apex have degree 2, apex has degree n − 1 and all other vertices have degree 3. Theorem 3.4. Shell graphs Sn,n−3 are D4-magic. Proof. Let us denote the vertices of the shell graph Sn,n−3 by u1, u2, . . . , un such that ui is adjacent to ui+1, where i = 1, 2, . . . , n and un+1 = u1. Without loss of generality let the apex be u1. Now consider the following cases: Case (i): n is even. We will define the map f : E(Sn,n−3) → N2n−3 as f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(unu1) = n, f(u1uj) = n + (j − 2), 3 ≤ j ≤ n − 1 174 D4 magic graphs and we define g : E(Sn,n−3) → D4 \ {ρ0} as g(u1u2) = g(unu1) = ρ2, g(u1ui) = µ1, 3 ≤ i ≤ n − 1, g(uiui+1) = µ2, 2 ≤ i ≤ n − 1. Clearly f and g define a D4-magic labeling with magic constant µ1. Case (ii): n is odd. Define f as f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(u1un) = n, f(u1uj) = n + (j − 2), 3 ≤ j ≤ n − 1 and define g as g(u1u2) = g(u1un) = ρ2, g(u1uj) = µ1, 3 ≤ j ≤ n − 1, g(uiui+1) = { ρ2, if i is even, µ2, if i is odd, 1 < i ≤ n − 1. Obviously the functions f and g define a D4-magic labeling of Sn,n−3 with magic constant ρ0. This completes the proof of the theorem. When k copies of Cn share a common edge it will form the n-gon book of k pages and is denoted by B(n, k). Theorem 3.5. The graph n-gon book of k pages B(n, k) is D4-magic. Proof. Let G be the graph B(n, k). Denote the vertices of common edge by k1 and kn and the edges of ith page other than k1 and kn by ui2, ui3, . . . , uin−1 such that ui2 is adjacent to k1 and uin−1 adjacent to kn and uij adjacent to uij+1 for all 2 ≤ j < n − 1. Consider the following cases: Case (i): k is even. Define g : E(G) → D4 \ {ρ0} as g(k1kn) = ρ2, g(u1ju1j+1) = µ1, 2 ≤ j ≤ n − 2, g(u1n−1kn) = µ1 = g(k1u12), g(uijuij+1) = µ2, 2 ≤ i ≤ k, 2 ≤ j ≤ n − 1, g(k1ul2) = g(uln−1) = µ2, 2 ≤ l ≤ k. 175 C. Anusha and V. Anil Kumar Now define f as f(k1kn) = 1, f(k1u12) = 2, f(u1n−1kn) = n, f(u1ju1j+1) = j + 1, ∀ 2 ≤ j ≤ n − 2, f(k1ui2) = n + (i − 2)(n − 1) + 1, i ≥ 2, f(uijuij+1) = n + (i − 2)(n − 1) + j, 2 ≤ j ≤ n − 2, 2 ≤ i ≤ k, f(ui(n−1)kn) = n + (i − 2)(n − 1) + (n − 1), 2 ≤ i ≤ k. The functions f and g determine a D4-magic labeling with magic constant ρ0. Case (ii): k is odd. Here define g as g(e) = ρ2, ∀e ∈ E(G) then g together with any bijection f : E(G) → Nkn−1 will define a D4-magic labeling of B(n, k) with magic constant ρ0. This completes the proof of the theorem. Note that, for any n ≥ 3 the path graph of order n is not D4-magic. 4 Path Generated Graphs In this section we will consider some graphs which are constructed from Paths. We start with the Splitting graph of Path. A splitting graph S(G) of a graph G is the graph obtained from G by adding to G a new vertex z′ for each vertex z of G and joining z′ to the neighbors of z in G. Theorem 4.1. Splitting graph of the path graph Pn, n ≥ 3 is D4-magic. Proof. Let Pn be a path graph of order n, where n ≥ 3. Let u1, u2, . . . , un be the vertices of Pn, where uiui+1 ∈ E(Pn), i = 1, 2, . . . , n − 1. There are 2n vertices and 3n − 3 edges in S(Pn). Let un+i be the vertex corresponding to the ith vertex in S(Pn). Observe that there are two pendant edges in S(Pn), one with end points u2 and un+1 and the other with end points un−1 and u2n. Case (i): n = 3. In this case, define f : E(S(P3)) → N6 as f(u1u2) = 1, f(u2u3) = 3, f(u3u5) = 2, f(u1u5) = 4, f(u2u4) = 5, f(u2u6) = 6. Now define g : E(G) → D4 \ {ρ0} as g(u1u2) = g(u3u5) = ρ1, g(u2u3) = g(u1u5) = δ2, g(u2u4) = g(u2u6) = µ1. 176 D4 magic graphs Case (ii): n > 3. In this case, define f and g as follows: f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(uiun+(i−1)) = n + (i − 2), 2 ≤ i ≤ n, f(uiun+(i+1)) = (2n − 2) + i, 1 ≤ i ≤ n − 1 and g(u1u2) = ρ2, g(un−1un) = µ2, g(u2un+1) = g(un−1u2n) = µ1, g(uiui+1) = µ1, 2 ≤ i < n − 1, g(uiun+(i−1)) = ρ2, 3 ≤ i ≤ n, g(uiun+(i+1)) = µ2, 1 ≤ i ≤ n − 2. In all the above cases, we can prove that the functions f and g defines a D4-magic labeling of S(Pn) with magic constant µ1. This completes the proof of the theorem. The middle graph of a connected graph G denoted by M(G) is the graph whose vertex set is V (G) ∪ E(G) where two vertices are adjacent if (i) They are adjacent edges of G or (ii) One is a vertex of G and the other is an edge incident with it. Theorem 4.2. Middle graph of the path graph Pn is D4-magic for n ≥ 3. Proof. Let M(Pn) be the middle graph of the path Pn. Denote the vertices of Pn by u1, u2, . . . , un and edges by e1, e2, . . . , en−1 where ei incident with ui and ui+1. There are 2n − 1 vertices and 3n − 4 edges in M(Pn). Consider the following cases: Case(i): n = 3. Define f : E(M(P3)) → N3n−4 as f(e1u1) = 1, f(e1u2) = 2, f(e1e2) = 3, f(e2u2) = 4 and f(e2u3) = 5 and define g : E(M(P3)) → D4 \ {ρ0} as g(e1u1) = ρ2 = g(e2u3), g(e1u2) = ρ1 = g(e2u2), g(e1e2) = ρ3. Then clearly the middle graph of the path P3 is D4-magic with magic constant ρ2. Case(ii): n > 3. Define f : E(M(Pn)) → N3n−4 as follows: For 1 ≤ i ≤ n − 2, 1 ≤ j ≤ n, f(eiuj) = 2(i − 1) + j and f(eiei+1) = 3i. 177 C. Anusha and V. Anil Kumar Now define g : E(M(Pn)) → D4 \ {ρ0 by g(e1u1) = ρ2 = g(en−1un), g(e1u2) = µ2, g(e2u2) = µ1, g(e2e3) = ρ3 g(e2u3) = ρ1 = g(e3u3), g(eiei+1) = µ2, where i ̸= 2 and 1 ≤ i ≤ n − 1, g(eiuj) = { µ1, if j = i + 1 and 3 ≤ i < n − 1, µ2, if i = j and 4 ≤ i ≤ n − 1. The above functions f and g will define a D4-magic labeling of M(Pn) with magic constant ρ2. This completes the proof of the theorem. A triangular snake Tn is obtained from the path Pn by replacing each edge of the path by a triangle C3. Theorem 4.3. The Triangular snake Tn is D4-magic. Proof. Note that every vertex of Tn has even degree . So the proof is indis- putable from Theorem 2.1. The alternate triangular snake A(Tn) is obtained from the path u1, u2, . . . , un by joining uiui+1(alternatively) to a new vertex vi. Theorem 4.4. The alternate triangular graph A(Tn) is D4-magic. Proof. Let us denote the vertices of the path Pn be u1, u2, . . . , un and the vertex which join ui and ui+1 be denoted by vi. Now consider the following cases: Case (i): n is even and triangle starts from u1. Suppose that n is even and the triangle starts from the first vertex ui, then there are n + n 2 vertices and 2n − 1 edges. Suppose n = 2 then A(Tn) is C3 itself. So there is nothing to prove. Suppose n = 4 then take f be any bijection from E(A(Tn)) to N7 and define g : E(A(Tn)) → D4 \ {ρ0} by g(u1v1) = g(u4v3) = ρ1, g(u2u3) = ρ2, g(u2v1) = g(u3v3) = g(u1u2) = g(u3u4) = ρ3. Then A(T4) becomes D4-magic with magic constant ρ0. Suppose n > 4, then let f : E(A(Tn)) → N2n−1 be any bijection and define 178 D4 magic graphs g : E(A(Tn)) → D4 \ {ρ0} as g(u1u2) = g(un−1un) = g(u2v1) = g(un−1vn−1) = ρ3, g(u1v1) = g(un−1vn−1) = ρ1. For 2 ≤ i < n − 1, define g(uiui+1) = { ρ2, if i is even, µ2, if i is odd. g(u2k+1v2k+1) = g(u2k+2v2k+1) = µ1, k = 1, 2, . . . , n − 4 2 Obviously the functions f and g will constitute a D4-magic labeling for A(Tn) with l∗(u) = ρ0, ∀u ∈ V (A(Tn)). Case (ii): n is even and the triangle starts from the second vertex u2. We can define a magic labeling for A(Tn), where n is even and cycle starts from u2 as follows: Let f be any bijection as above and define g as g(uiui+1) = { ρ2, if i is odd, ρ3, if i is even, , 1 ≤ i ≤ n − 1, g(u2kv2k) = g(u2k+1v2k) = ρ1, k = 1, 2, 3, . . . , n − 2 2 . Clearly l∗ is a constant map, i.e., l∗(u) = ρ2, ∀u ∈ V (A(Tn)). Case (iii): n is odd and the triangle starts from the first vertex. Suppose n = 3 and the triangle starts from the first vertex u1. Let f : E(A(Tn)) → N4 be any bijection. Now define g : E(A(Tn)) → D4 \ {ρ0} by g(u1u2) = g(u2v1) = δ2, g(u1v1) = δ1, g(u2u3) = ρ2. Using these maps we can show that the graph is D4-magic with magic constant ρ2. When n is odd and n > 3, there are n+ (n−1) 2 vertices and 2(n−1) edges in A(Tn). Suppose that n > 3, n is odd and the triangle of A(Tn) starts from the first vertex u1. Here we take f as any bijection and g : E(A(Tn)) → D4 \ {ρ0} be defined as follows: g(u1u2) = g(u2v1) = δ2, g(u1v1) = δ1, g(uiui+1) = { ρ2, if i is even, ρ3, if i is odd, , 1 < i < n. g(uivi) = g(ui+1vi) = ρ1, 1 < i < n and i is odd. Then clearly l∗(u) = ρ2, ∀v ∈ V (A(Tn)) 179 C. Anusha and V. Anil Kumar Case (iv): n is odd and triangle starts from the second vertex. A(Tn) with n odd and the triangle starts from the first vertex is just the mir- ror image of the A(Tn) in Case (iii). So we can define f and g similarly as in Case(iii) and obtain a D4-magic labeling for A(Tn) with magic constant ρ2. This completes the proof of the theorem. A double triangular snake D(Tn) consists of two triangular snakes that have a common path Pn. Theorem 4.5. The double triangular graph is D4-magic. Proof. Let u1, u2, . . . , un be the vertices of the path Pn and let v1, v2, . . . , vn−1, w1, w2, . . . , wn−1 be the remaining vertices of D(Tn) such that the vertex vi is ad- jacent to ui and ui+1, where 1 ≤ i < n. Similarly the vertex wi is adjacent to ui and ui+1. Without loss of generality let v1, v2, . . . , vn−1 and w1, w2, . . . , wn−1 be the vertices of upper triangles and lower triangles respectively. Now we define a D4-magic labeling for D(Tn) as follows: Let f : E(D(Tn)) → N5(n−1) be any bijection and let g : E(D(Tn)) → D4 \ {ρ0} be defined by g(uiui+1) = ρ2, g(uivi) = g(uiwi) = ρ1, and g(ui+1vi) = g(ui+1wi) = ρ3 where 1 ≤ i < n. Thus we can see that l∗(u) = ρ0, ∀u ∈ V (D(Tn)). This completes the proof. 5 Conclusions In this paper, we introduced the concept of A-magic labeling of graphs, where A is a nonabelian group. 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