On (F, H)-simultaneously-magic labelings of graphs

Electronic Journal of Graph Theory and Applications 11 (1) (2023), 49–64 On (F, H)-sim-magic labelings of graphs Yeva Fadhilah Asharia , A.N.M. Salmana,b , Rinovia Simanjuntak∗,a,b , Andrea Semaničová-Feňovčı́kovác , Martin Bačac Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia b Center for Research Collaboration on Graph Theory and Combinatorics, Indonesia c Department of Applied Mathematics and Informatics, Technical University in Košice, Slovakia a [email protected], [email protected], [email protected], [email protected], [email protected] ∗ corresponding author Abstract A simple graph G(V, E) admits an H-covering if every edge in G belongs to a subgraph of G isomorphic to H. In this case, G is called H-magic if there exists a bijective function f : V ∪ E → {1, 2, . . . , |VP| + |E|}, such that for every subgraph H ′ of G isomorphic to H, wtf (H ′ ) = P e∈E(H ′ ) f (e) is constant. Moreover, G is called H-supermagic if f : V (G) → v∈V (H ′ ) f (v) + {1, 2, . . . , |V |}. This paper generalizes the previous labeling by introducing the (F, H)-sim-(super) magic labeling. A graph admitting an F -covering and an H-covering is called (F, H)-sim-(super) magic if there exists a function f that is F -(super)magic and H-(super)magic at the same time. We consider such labelings for two product graphs: the join product and the Cartesian product. In particular, we establish a sufficient condition for the join product G + H to be (K2 + H, 2K2 + H)sim-supermagic and show that the Cartesian product G × K2 is (C4 , H)-sim-supermagic, for H isomorphic to a ladder or an even cycle. Moreover, we also present a connection between an α-labeling of a tree T and a (C4 , C6 )-sim-supermagic labeling of the Cartesian product T × K2 . Keywords: H-covering, H-(super)magic, (F, H)-sim-(super)magic, join product, Cartesian product Mathematics Subject Classification: 05C70, 05C76, 05C78 DOI: 10.5614/ejgta.2023.11.1.5 Received: 2 July 2022, Revised: 6 January 2023, Accepted: 4 February 2023. 49 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. 1. Introduction The graphs considered in this paper are finite and simple. Let G be a graph, with the vertex set V (G) and the edge set E(G). The cardinalities of V (G) and E(G) are called the order and the size of G, respectively. A labeling f of G is a map that assigns certain elements of G to positive or non-negative integers. In this paper, we consider a total labeling of G as a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| P + |E(G)|}. Under a total labeling f , the weight of a vertex v ∈ V (G) is wtf (v) = f (v) + vw∈E(G) f (vw) and the weight of an edge vw ∈ E(G) is wtf (vw) = f (v) + f (vw) + f (w). Simanjuntak et al. [27] introduced an (a, d)-edge-antimagic total labeling ((a, d)-EAT) as a total labeling f where the set of edge-weights {wtf (vw)|vw ∈ E(G)} constitutes a set of an arithmetic progression {a, a + d, . . . , a + (|E(G)| − 1)d} for two integers a > 0 and d ≥ 0. When d = 0, the (a, 0)-edge(vertex)-antimagic labeling was previously known as the edge-magic total labeling (EMT) and was introduced by Kotzig and Rosa [15] in 1970. When G has EMT or (a, d)-EAT labelings and the corresponding f labeling has the property f (V (G)) = {1, 2, . . . , |V (G)|}, we say that G is super edge-magic total (SEMT) or super (a, d)-edge-antimagic total ((a, d)-SEAT), respectively. Another variation of magic labeling called vertex-magic total labeling was introduced by MacDougal et al. [17]. A vertex-magic total labeling (VMT) of G is a total labeling where there exists a positive integer k such that the vertex-weight wtf (v) = k for every vertex v of G. If {wtf (v)|v ∈ V (G)} = {a, a + d, . . . , a + (|V (G)| − 1)d} for two integers a > 0 and d ≥ 0, the labeling f of G is called (a, d)-vertex-antimagic total labeling ((a, d)-VAT), that was first introduced by Bača et al. [3]. Comprehensive surveys about the existence of magic and antimagic graphs can be found in [4, 5, 11, 29]. In 2005, as an extension of the edge-magic total labeling, Gutiérez and Lladó [12] introduced an H-magic labeling of a graph. A graph G admits an H-covering if every edge in E(G) belongs to a subgraph of G isomorphic to a given graph H. A total labeling f of G is an PH-magic labeling P f (v) + if there exists a positive integer k such that wt(H ′ ) = e∈E(H ′ ) f (e) = k v∈V (H ′ ) ′ for every subgraph H of G isomorphic to H. In this case, G is called an H-magic graph. If f (V ) = {1, 2, . . . , |V (G)|}, then G is said to be an H-supermagic graph. Current results on H-magic labelings can be seen in the survey [11]. In 2005, Exoo et al. [9] asked whether there exists a labeling of a graph that is simultaneously vertex-magic and edge-magic and called such labeling totally magic. Subsequently, in 2005, Bača et al. [6] extended a similar question for (a, d)-EAT labeling and (a, d)-VAT labelings; and defined the totally antimagic total (TAT) labeling. Motivated by the two notions above, it is interesting to ask a similar question by considering the subgraph covering in G. Suppose that G simultaneously admits an F -covering and an H-covering. We propose a new notion of a labeling called an (F, H)-sim-magic labeling as a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| + |E(G)|} where there exist two positive integers kF and kH (not necessarily the same) such that X X wtf (F ′ ) = f (v) + f (e) = kF v∈V (F ′ ) e∈E(F ′ ) 50 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. and wtf (H ′ ) = X f (v) + v∈V (H ′ ) X f (e) = kH , e∈E(H ′ ) for each subgraph F ′ of G isomorphic to F and each subgraph H ′ of G isomorphic to H. We say that G is (F, H)-sim-magic. Furthermore, if f (V (G)) = {1, 2, . . . , |V (G)|}, G is said to be (F, H)-sim-supermagic. The simplest example of a (F, H)-sim-magic graph can be deduced from previously known H-magic labelings. For odd m and n at least three, the disjoint union of m cycles mCn is both SEMT [10] and Cn -supermagic [1, 18]. Although the Cn -supermagic labelings described in [1, 18] are not SEMT, the SEMT labeling of 3C3 described in [10] is also C3 -supermagic (see Figure 1). This implies that 3C3 is (K2 , C3 )-sim-supermagic. Figure 1. A (K2 , C3 )-sim-supermagic graph. An interesting fact for (F, H)-sim-magic labeling is that although a graph is both F -magic and H-magic, such a graph does not need to be (F, H)-sim-magic. An example is the fan Fn with vertex-set V (Fn ) = {vi |0 ≤ i ≤ n} and edge-set E(Fn ) = {vi vi+1 |1 ≤ i ≤ n − 1} ∪ {v0 vi |1 ≤ i ≤ n}. It is known that, for every n ≥ 3, Fn is EMT (see [28]) and C3 -supermagic (see [21]). However, for every n ≥ 3, Fn is not (K2 , C3 )-sim-magic as stated in the following theorem. Theorem 1.1. Let n ≥ 3 be a positive integer. A fan Fn is not (K2 , C3 )-sim-magic. Proof. Suppose that Fn is a (K2 , C3 )-sim-magic graph and let f be a (K2 , C3 )-sim-magic labeling of Fn with a magic constant pair (k1 , k2 ). Consider the weights of two C3 cycles v0 v1 , v1 v2 , v2 v0 and v0 v2 , v2 v3 , v3 v0 . As these weights are equal, we have 2 X f (vi ) + f (v0 v1 ) + f (v1 v2 ) + f (v2 v0 ) = i=0 3 X f (vi ) + f (v0 v2 ) + f (v2 v3 ) + f (v3 v0 ), i=1 and so f (v1 ) + f (v1 v0 ) + f (v1 v2 ) = f (v3 ) + f (v2 v3 ) + f (v0 v3 ). (1) Adding f (v0 ) to both sides of Equation (1) and using the fact that all edges have the same edge weight, we obtain f (v1 v2 ) = f (v2 v3 ), a contradiction. In this paper, we study simultaneous labelings for two product graphs: the join product and Cartesian product graphs. In particular, we investigate a sufficient condition for the join product 51 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. graph G + H to be (K2 + H, 2K2 + H)-sim-supermagic (Section 3). We construct (C4 , H)-simsupermagic labelings for the Cartesian product G × K2 , where H is isomorphic to a ladder or an even cycle (Section 4). Finally, in the last section, we provide relationships between an α-labeling of a tree T and a (C4 , C6 )-sim-supermagic labeling of the Cartesian product T × K2 . Throughout the paper, we shall use the following definitions and notations. The degree of a vertex v is denoted by deg(v). For a connected graph H, a graph G is H-free if G does not contain H as a subgraph. Notations for some classes of graphs can be seen in Table 1. Table 1. Classes of graphs Notation Cn Kn K1,n Pn Sn1 ,n2 ,...,nk Notes A cycle on n vertices, n ≥ 3. A complete graph on n vertices, n ≥ 1. A star with one internal vertex and n leaves, n ≥ 2. A path on n vertices, n ≥ 2. A caterpillar is a graph derived from a path Pk , k ≥ 2, where for i ∈ {1, 2, ..., k}, each vi ∈ V (Pk ) is adjacent to ni ≥ 0 additional leaves. 2. Balanced and Anti Balanced Multisets A multiset is a generalization of a set where repetition of elements is allowed. Let a and b be two integers. We use the notation [a, b] to define the set of consecutive integers {a, a + 1, . . . , b}. So [a, b] = ∅, if a > b. For an integer k, thePaddition k + [a, b] means [a + k, b + k] and for a P multiset of integers Y , we denote x∈Y x by Y . Let x be an element of a multiset Y . Then, the multiplicity of x, denoted by U mY (x), is the number of occurrences of x in Y . Let X and Y be two multisets. AU multiset sum X Y is a union of X and Y , where mX U Y (x) U = mX (x) + mY (x) for each x ∈ X Y . For example, if X = {a} and Y = {a, a, b}, then, X Y = {a, a, a, b}. We shall utilize the notions of a k-balanced partition of a multiset introduced by Maryati et al. [19] and a (k, δ)-anti balanced partition of a multiset introduced by Inayah et al. [13] to construct labelings in Sections 3 and 4. Let k and δ be two positive integers, and X be a multiset containing positive integers. X is said to be (k, δ)-anti balanced if there exist k subsets of X, say U , ki=1 Xi = X, and for each i ∈ [1, k−1], X 1 , X2 , . . . P , Xk , such that for every i ∈ [1, k], |Xi | = |X| k P Xi+1 − Xi = δ. For every i ∈ [1, k], Xi is called P a (k, δ)-anti balanced subset of X. In the case that there exists a positive integer θ such that Xi = θ for every i ∈ [1, k], then X is called k-balanced with Xi s as k-balanced subsets of X. Lemma 2.1. [18] Let x, y, and k be three integers, where 1 ≤ x < y and k > 1. If X = [x, y] and P (x + y) for every i ∈ [1, k]. |X| is a multiple of 2k, then X is k-balanced with Xi = |X| 2k Lemma 2.2. Let x and k be positive integers, k ≥ 2. If ( [x, x + 2k − 1], for odd k; X= k [x, x + 2k] \ {x + 2 }, for even k,   P then X is (k, 1)-anti balanced with Xi = 2x + i + 3 k2 for every i ∈ [1, k]. 52 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al.     + 2(1 − (i mod 2)) k2 Proof. For each i ∈ [1, k], define Xi = {ai , bi }, where ai = x − 1 + i+1 2 k  i  k Uk i and b = x + + + 2(i mod 2) . Thus, i=1 X 2 2 2   P P Pi = X and we have |Xi | = 2 and k Xi = 2x + i + 3 2 for each i ∈ [1, k]. Since Xi+1 − Xi = 1 for every i ∈ [1, k − 1], X is a (k, 1)-anti balanced. Lemma 2.3. Let Px and k be positive integers, k ≥ 2. If X = [x, x + 2k − 1], then X is (k, 2)-anti balanced with Xi = 2(x + i − 1) + k for every i ∈ [1, k]. Uk Proof. P Define Xi = {x − 1 + i, x + i + k − 1} for each i ∈ [1, k]. Hence, i=1 Xi = X, |Xi | = 2, and Xi = P P 2(x + i − 1) + k for every i ∈ [1, k]. We have that X is (k, 2)-anti balanced since Xi+1 − Xi = 2 for every i ∈ [1, k − 1]. 3. Labelings for Join Product Graphs Let G ∪ H denote the disjoint union of G and H. Then, the join product G + H of two disjoint graphs G and H is the graph G ∪ H together with all the edges joining vertices of G and vertices of H. The study of H-magicness of join product graphs has been conducted for some particular families of graphs, as summarized in Table 2. Table 2. Known join product graphs which are H-magic Join product P n + K1 , n ≥ 3 Cn + K1 , n odd, n ≥ 5 n even, n ≥ 4 C n + K1 , n ≥ 3 K1,n + K1 , n ≥ 3 nK2 + K1 , n ≥ 2 H C3 C4 C3 C3 C4 C3 C3 Reference Ngurah et al. [21] and Ovais et al. [22] Ovais et al. [22] Lladó and Moragas [16] Roswitha et al. [25] Semaničová-Feňovčı́ková et al. [26] Ngurah et al. [21] Lladó and Moragas [16] The following theorem provides a sufficient condition for the join product graph G + H to be (K2 + H, 2K2 + H)-sim-supermagic. Theorem 3.1. Let G and H be two connected graphs such that G admits a 2K2 -covering and G + H contains exactly |E(G)| subgraphs isomorphic to K2 + H. If G is SEMT, then G + H is (K2 + H, 2K2 + H)-sim-supermagic. Proof. Let g be a super edge-magic total (SEMT) labeling of G with the magic constant mg . Let V (G) = {vi |vi = g −1 (i) and i ∈ [1, p]}, V (H) = {ui |i ∈ [1, r]}, E(G) = {ei |i ∈ [1, q]}, and E(H) = {ki |i ∈ [1, s]}. Hence, |V (G)| = p, |E(G)| = q, |V (H)| = r, |E(H)| = s. Thus, V (G + H) = V (G) ∪ V (H) and E(G + H) = E(G) ∪ E(H) ∪ {ui vj |i ∈ [1, r] and j ∈ [1, p]}. Consider Y = [1, p + q + r + s + p r] as the set of all labels of vertices and edges in G + H. Then, we divide the proof into three cases. Case 1. r is even. Partition Y into five subsets, namely A = [1, r], B = r +[1, p], C = (r +p)+[1, p r], D = (r +p+ 53 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. p r)+[1, q] and E = (r+p+p r+q)+[1, r is even, |C| is a multiple of 2p. By Lemma 2.1, P s]. Since pr we have that C is p-balanced with Ci = 2p (r + p + 1 + r + p + pr) = 21 r(p(r + 2) + 2r + 1) for each i ∈ [1, p]. Next, label the vertices and edges in G+H by total labeling f as defined in the following steps. For each i ∈ [1, r], f (ui ) = i. For each i ∈ [1, p], f (vi ) = i + r. For each j ∈ [1, p] and i ∈ [1, r], f (ui vj ) = mi , where mi ∈ Cj . For each ei ∈ E(G) and i ∈ [1, q], f (ei ) = g(ei ) + r + pr. For each ki ∈ E(H) and i ∈ [1, s], f (ki ) = mi , where mi ∈ E and no two distinct edges in E(H) are assigned the same number. S S S Thus, we get ri=1 {f (ui )} = A, pi=1 {f (vi )} = B, pj=1 {f (ui vj )|i ∈ [1, r]} = C, {f (vi vj )|vi vj ∈ E(G)} = D, and {f (ui uj )|ui uj ∈ E(H)} = E. Clearly, f is a bijective function from V (G + H) ∪ E(G + H) to Y . Let F be a subgraph of G + H isomorphic to K2 + H. It is clear that F contains exactly one edge of E(G), say vx vy for some distinct x, y ∈ [1, p]. Then, V (F ) = V (H) ∪ {vx , vy } and E(F ) = E(H) ∪ {vx vy } ∪ {ui vj |j ∈ {x, y}, i ∈ [1, r]}. Thus, 1. 2. 3. 4. 5. wtf (F ) = r X f (ui ) + f (vx ) + f (vy ) + i=1 X f (e) + f (vx vy ) + [f (ui vx ) + f (ui vy )] i=1 e∈E(H) = [g(vx ) + g(vy ) + g(vx vy )] + 3r + pr + r X 1 r(r 2 + 1) + s(r + p + pr + q) + s X i i=1 +r(p(r + 2) + 2r + 1). Since [g(vx ) + g(vy ) + g(vx vy )] = mg , we see that wtf (F ) is independent of F . Now, let F ′ be a subgraph of G + H isomorphic to 2K2 + H. It is clear that F ′ contains  two P P non-adjacent edges of E(G). Then, wtf (F ′ ) = 2wtf (F ) − e∈E(H) f (e) . So, u∈V (H) f (u) + ′ ′ wtf (F ) is independent of F . Case 2. r is odd and p is odd. Partition Y into five subsets, namely A = [1, r], B = r + [1, 2p], C = (r + 2p) + [1, p(r − 1)], D = (r + p +P p r) + [1, q], and E = (r + p + p r + q) + [1, s]. By Lemma 2.2, B is (p, 1)-anti balanced with Bi = 2(r + 1) + 3 p2 +Pi for every i ∈ [1, p]. Since g is an injective function, g −1 (i) = vi for every i ∈ [1, p]. This gives Bi = 2(r + 1) + 3⌊ p2 ⌋ + i = 2(r + 1) + 3⌊ p2 ⌋ + g(vi ) for every i ∈ [1, p]. The cardinality of C is a multiple of 2p. By Lemma 2.1, C is p-balanced with P (r + 2p + 1 + r + 2p + p(r − 1)) = 12 (r − 1)(2r + 3p + pr + 1) for every i ∈ [1, p]. Ci = p(r−1) 2p Next, label the vertices and edges in G+H by total labeling f as defined in the following steps. 1. For each i ∈ [1, r], f (ui ) = i. 2. For each i ∈ [1, p], f (vi ) = min{x|x ∈ Bi }. 3. For each i ∈ [1, p], f (u1 vi ) = bi , where bi ∈ Bi \ {f (vi )}. 54 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. 4. For each j ∈ [1, p] and i ∈ [2, r], f (ui vj ) = mi , where mi ∈ Cj . 5. For each ei ∈ E(G) and i ∈ [1, q], f (ei ) = g(ei ) + r + pr. 6. For each ki ∈ E(H) and i ∈ [1, s], ki = mi , where mi ∈ E and no two distinct edges are assigned the same number. S S S Thus, we get ri=1 {f (ui )} = A, pi=1 {f (vi ), f (u1 vi )} = B, pj=1 {f (ui vj )| i ∈ [2, r]} = C, {f (vi vj )|vi vj ∈ E(G)} = D, and {f (ui uj )|ui uj ∈ E(H)} = E. Clearly, f is a bijective function from V (G + H) ∪ E(G + H) to Y . Let F be a subgraph of G + H isomorphic to K2 + H. It is clear that F contains exactly one edge of E(G), say vx vy for some distinct x, y ∈ [1, p]. Then, V (F ) = V (H) ∪ {vx , vy } and E(F ) = E(H) ∪ {vx vy } ∪ {ui vj |j ∈ {x, y}, i ∈ [1, r]}. Thus, wtf (F ) = r X i=1 f (ui ) + f (vx ) + f (vy ) + X e∈E(H) f (e) + f (vx vy ) + r X [f (ui vx ) + f (ui vy )] i=1   = [g(vx ) + g(vy ) + g(vx vy )] + 4(r + 1) + 6 p2 + r + pr + 21 r(r + 1) +s(r + p + pr + q) + 12 s(s + 1) + (r − 1)(2r + 3p + pr + 1). Since [g(vx ) + g(vy ) + g(vx vy )] = mg , we see that wtf (F ) is independent on the choosing of F . Now, let F ′ be a subgraph of G + H isomorphic to 2K2 + H. It is clear that F ′ contains  two P P ′ non-adjacent edges of E(G). Thus, wtf (F ) = 2wtf (F ) − e∈E(H) f (e) . So, v∈V (H) f (v) + wtf (F ′ ) is independent of F ′ . Case 3. r is odd and p is even. Partition Y into five subsets, A = [1, r − 1] ∪ {r + p2 }, B = [r, r + 2p] \ {r + p2 }, C = (r + 2p) + [1, p (r − 1)], D = (r + pP + p r) + [1, q], and E = (r + p + pr + q) + [1, s]. By Lemma 2.2, B is p ⌋ for each i ∈ [1, p]. Since g is an injective function, (p, 1)-anti balanced with Bi = 2r + i + 3⌊ 2 P −1 g (i) = vi for every i ∈ [1, p]. Therefore, Bi = 2r + 3⌊ p2 ⌋ + i = 2r + 3⌊ p2 ⌋ + g(vi ) for every i ∈ [1, p]. Now, of C is a multiple of 2p. By Lemma 2.1, we have that C is p-balanced P the cardinality p(r−1) with Ci = 2p (r + 2p + 1 + r + 2p + pr) = 21 (r − 1)(2r + 3p + pr + 1) for every i ∈ [1, p]. Next, label the vertices and edges in G + H by the total labeling f defined in the following steps. For each i ∈ [2, r], f (ui ) = i − 1, and f (u1 ) = r + p2 . For each i ∈ [1, p], f (vi ) = min{x|x ∈ Bi }. For each i ∈ [1, p], f (u1 vi ) = bi , where bi ∈ Bi \ {f (vi )}. For each j ∈ [1, p] and i ∈ [2, r], f (ui vj ) = mi , where mi ∈ Cj . For each ei ∈ E(G) and i ∈ [1, q], f (ei ) = g(ei ) + r + pr. For each ki ∈ E(H) and i ∈ [1, s], f (ki ) = mi , where mi ∈ E and no two distinct edges are assigned the same number. S S S Then, ri=1 {f (ui )} = A, pi=1 {f (vi ), f (u1 vi )} = B, pj=1 {f (ui vj )|i ∈ [2, r]} = C, {f (vi vj )|vi vj ∈ E(G + H)} = D and {f (ui uj )|ui uj ∈ E(G + H)} = E. Clearly, f is a bijective function from V (G + H) ∪ E(G + H) to Y . 1. 2. 3. 4. 5. 6. 55 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. Let F be a subgraph of G + H isomorphic to K2 + H. Then F contains exactly one edge of E(G), say vx vy for some distinct x, y ∈ [1, p]. Then, F has the form V (F ) = V (H) ∪ {vx , vy } and E(F ) = E(H) ∪ {vx vy } ∪ {ui vj |j ∈ {x, y}, i ∈ [1, r]}. Thus, wtf (F ) = r X f (ui ) + f (vx ) + f (vy ) + i=1 = [g(vx ) + g(vy ) + g(vx vy )] + X e∈E(H) r X i+ i=2 +2r2 + rs + qs + 4r. f (e) + f (vx vy ) + r X f (ui vx ) + i=1 s X r X f (ui vy ) i=1 i + p r2 + r(s + 3) + s − i=1 5 2
+6 p 2 Since [g(vx ) + g(vy ) + g(vx vy )] = mg , we see that wtf (F ) is independent on the choosing of F . ′ Now, let F ′ be a subgraph of G + H isomorphic P to 2K2 + H.PF contains two non-adjacent ′ edges of E(G). Thus, wtf (F ′ ) = 2wtf (F ) − e∈E(H) f (e) . So, wtf (F ) is v∈V (H) f (v) + independent of F ′ . An example of the labeling depicted in the proof of Theorem 3.1 can be seen in Figure 2 where a (K5 , 2K2 + C3 )-sim-supermagic labeling of S2,0,0,2 + C3 is presented. Figure 2. A (K5 , 2K2 + C3 )-sim-supermagic labeling of S2,0,0,2 + C3 The following corollary is a consequence of Theorem 3.1 with H = K1 . Corollary 3.1. Let G be a C3 -free connected graph containing a P5 . If G is SEMT graph, then G + K1 is (C3 , 2K2 + K1 )-sim-supermagic. This corollary enlarges the classes of graphs known to be C3 -supermagic; since up to date, only the following join product graphs were known to be C3 -supermagic: Pn +K1 , Cn +K1 , K1,n +K1 , and nK2 + K1 , where n ≥ 3 [16, 21, 22, 25]. 56 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. 4. Labelings for Cartesian Product Graphs The Cartesian product of two graphs G and H, denoted by G × H, is a graph whose vertex set is V (G) × V (H) = {(u, v)|u ∈ V (G), u ∈ V (H)} and for which two vertices (u1 , v1 ) and (u2 , v2 ) are adjacent if and only if either u1 u2 ∈ E(G) and v1 = v2 or v1 v2 ∈ E(H) and u1 = u2 . In this section, we shall study (F, H)-sim-supermagic labeling for the Cartesian product of an arbitrary graph G with K2 . The following notations are used or vertices and edges in G × K2 For each x ∈ V (G), let x and x′ be the corresponding vertices in the two copies of G in G × K2 , and so xx′ ∈ E(G × K2 ). For each xy ∈ E(G), denote by xy and x′ y ′ the corresponding edges in the two copies of G in G × K2 . We summarize the Cartesian product graphs G × K2 known to be H-magic in Table 3. Table 3. G × K2 that are H-magic Cartesian product G × K2 P m × K2 mK1,n × K2 s(Pn+1 × K2 ) ∪ k(Pn × K2 ) m(Pn × K2 ) P n × K2 P n × K2 G × K2 H C4 C4 C4 C4 C4 C2m P m × K2 C4 (2G) × K2 C4 Conditions and Reference G is C4 -free and SEMT of odd size [16] m ≥ 3 [21] m ≥ 2 and n ≥ 1 [1] s ≥ 1, k ≥ 1 and n ≥ 2 [1] m ≥ 2 and n ≥ 2 [23] n ≥ 4 and m ∈ [3, ⌊ n2 ⌋ + 1] [20] n ≥ 4 and m ∈ [3, n − 1] [20] G is C4 -free, SEMT and a connected (p, q)-graph where p or q is odd [14] G is C4 -free, connected, bipartite (with partite sets U and V ) and G has a SEMT labeling f such that f (U ) = [1, |U |] [14] In [20], Ngurah et al. constructed (Pm × K2 )-supermagic labelings of the ladder Pn × K2 for every m ∈ [3, n − 1]. A more general result by Baca et al. [7] established the following sufficient conditions for the Cartesian product G1 × G2 to be (H × G2 )-supermagic as stated in the following theorem. On the other hand, in [14] and [16] it was proved that if G is connected of odd order or size, C4 -free, and SEMT, then G × K2 admits a C4 -supermagic labeling. Theorem 4.1. [7] Let G1 be a graph of odd order p1 ≥ 3 admitting an H-covering given by t subgraphs isomorphic to H. If G2 is a graph of even order q2 ≥ 2 and odd size p2 ≥ 3 and the graph G1 × G2 contains exactly t subgraphs isomorphic to H × G2 , then G1 × G2 is (H × G2 )supermagic. In the next theorem, we enlarge the classes of graphs known to be (Pm × K2 )-supermagic [20] and extend sufficient conditions for the existence of a C4 -supermagic labeling of G × K2 [14, 16] without considering a SEMT labeling of G. Furthermore, our result settles the remaining cases of Theorem 4.1 for p2 = 1 and q2 = 2. Theorem 4.2. Let G be a C4 -free connected graph of odd order p ≥ 5. If G admits a Pm -covering for some m ∈ [3, p − 1], then G × K2 is (C4 , Pm × K2 )-sim-supermagic. 57 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. Proof. Let p and q be the order and the size of G, respectively. Consider A = [1, 3p + 2q] as the set of integers used to label vertices and edges in G × P2 . Now, partition A into three sets W = [1, 2p], X = [2p + 1, P 3p], and Y = [3p +1,3p + 2q]. Since p is odd, by Lemma 2.2, W is (p, 1)-anti balanced with Wi = 2 + i + 3 p2 for every i ∈ [1, p]. Now, since |Y | = 2q, P Lemma 2.1 ensures that Y is q-balanced with Yj = 2q (3p + 1 + 3p + 2q) = 6p + 2q + 1 for 2q each j ∈ [1, q]. Let g and h be bijections from V (G) to [1, p] and from E(G) to [1, q], respectively. Next, define a total labeling f of G×K2 . For each x ∈ V (G), label x and x′ in G×K2 by the elements of Wg(x) chosen so that f (x) < f (x′ ) and define f (xx′ ) = 3p − g(x) + 1. ForSeach xy ∈ E(G), define f as a bijection from {xy, x′ y ′ } to Yh(xy) with f (xy) < f (x′ y ′ ). Hence, v∈V (G×K2 ) {f (v)} = W and S e∈E(G×K2 ) {f (e)} = X ∪Y . Consequently, f is a bijective function from V (G×K2 )∪E(G×K2 ) to A. Since G is C4 -free, there are q subgraphs of G × K2 isomorphic to C4 . Let F be a subgraph of G × K2 isomorphic to C4 . Then, V (F ) = {x, x′ , y, y ′ } and E(F ) = {xx′ , yy ′ , xy, x′ y ′ }, where x, y ∈ V (G) and xy ∈ E(G). Therefore, wtf (F ) = f (x) + f (x′ ) + f (y) + f (y ′ ) + f (xx′ ) + f (yy ′ ) + f (xy) + f (x′ y ′ ) X X X = Wg(x) + Wg(y) + 3p − g(x) + 1 + 3p − g(y) + 1 + Yh(xy) p = 12p + 6 2 + 2q + 7, which is independent of F . Moreover, as G admits a Pm -covering for some m ∈ [3, p − 1], we have that G × K2 admits a (Pm × K2 )-covering. Let H = x1 x2 . . . xm be a subgraph of G isomorphic to Pm . For each H, denote by H ′ = x′1 x′2 . . . x′m the corresponding subgraph in G′ . Thus, for each H, we obtain H ′′ with V (H ′′ ) = {x1 , x2 , . . . , xm , x′1 , x′2 , . . . , x′m } and E(H ′′ ) = E(H) ∪ E(H ′ ) ∪ {xx′ |x ∈ V (H)} as the corresponding subgraph in G × K2 isomorphic to Pm × K2 . We can verify that there are exactly t subgraphs of G × K2 isomorphic to Pm × K2 , where t is the number of subgraphs isomorphic to Pm in G. Thus, X X X X X wtf (H ′′ ) = f (v) + f (v) + f (e) + f (e) + f (vv ′ ) v∈V (H) = X v∈V (H ′ ) ′ [f (v) + f (v )] + v∈V (H) = =3m p 2 ′ [f (e) + f (e )] + X v∈V (H) [3p − g(v) + 1] v∈V (H) e∈E(H) X hX v∈V (H) e∈E(H ′ ) e∈E(H) X i i X X hX [3p − g(v) + 1] Yh(e) + Wg(v) + v∈V (H) e∈E(H) + 4m + 9mp + 2mq − 6p − 2q − 1, which is independent of H ′′ . Hence, G × K2 is (C4 , Pm × K2 )-sim-supermagic. An example of the labeling in the proof of Theorem 4.2 is depicted in Figure 3. [20], Ngurah et al. showed that the ladder Pn × K2 is C2m -supermagic for every m ∈ In n [3, 2 + 1]. Then it is natural to ask for which graphs G, the Cartesian product G × K2 is 58 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. Figure 3. A (C4 , Pm × K2 )-sim-supermagic labeling of C7 × K2 for every m ∈ [3, 6].     (C2x , C2y )-sim-supermagic for some x, y ∈ 3, n2 + 1 . We will answer this question in Theorem 4.3, but to do so, we need to recall the following notion that was first introduced by Simanjuntak et al. [27]. An injective function f from V (G) onto the set {1, 2, . . . , |V (G)|} is called (a, d)-edgeantimagic vertex labeling ((a, d)-EAV) if the set of edge-weights {w(xy) = f (x) + f (y)|xy ∈ E(G)} = {a, a + d, . . . , a + (|E(G)| − 1)d}, where a > 0 and d ≥ 0 are two integers. A graph G is said to be an (a, d)-edge-antimagic vertex ((a, d)-EAV) graph if G has an (a, d)-EAV labeling. In [4], it was shown that a connected graph G that is not a tree has no (a, d)-EAV labeling for d ̸= 1. Lemma 4.1. [4] Let G be a connected graph that is not a tree. If G has an (a, d)-EAV labeling, then d = 1. The next theorem describes a construction of a (C2x , C2y )-sim-supermagic labeling of G × K2 from an (a, 2)-EAV labeling for some x, y ∈ 3, n2 + 1 . Due to Lemma 4.1, we restrict our consideration to trees. Theorem 4.3. Let m, n and p be positive integers where 3 ≤ m < p. Let G be a tree on p vertices where p ≥ 5, such that G admits a Pm -covering for some m ∈ 3, n2 + 1 . If G is an (a, 2)-EAV graph, then G × K2 is (C2x , C2y )-sim-supermagic for all x, y ∈ [2, m]. Proof. Let p and q be the order and the size of G, respectively. Let g : V (G) → {1, 2, . . . , p} be an (a, 2)-EAV labeling of G. Since |V (G × K2 )| = 2p and |E(G × K2 )| = p + 2q, the set of labels used to label vertices and edges of G × K2 is A = [1, 3p + 2q]. Now, partition A into three sets W = [1, 2p], P X = [2p + 1, 3p] and Y = [3p + 1, 3p + 2q]. By Lemma 2.3, W is (p, 2)-anti balanced with Wi = 2i + p for P every i ∈ [1, p]. According to Lemma 2.3, Y is (q, 2)-anti balanced with Yj = 6p + q + 2j for each j ∈ [1, q]. Next, define a total labeling f of G × K2 . For each x ∈ V (G), label the corresponding vertices x and x′ in G × K2 by the elements of Wg(x) chosen so that f (x) < f (x′ ). For each x ∈ V (G), define f (xx′ ) = 3p + 1 − g(x). Now, for each xy ∈ E(G), label the corresponding edges xy and 59 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. x′ y ′ in G×K2 by the elements of Yr where r = 12 (2q +a−g(x)−g(y)) such that f (xy) < f (x′ y ′ ). It follows easily that f depends on g. Then, f is a bijective function from V (G × K2 ) ∪ E(G × K2 ) to A.     Since G admits a Pm -covering for some m ∈ 3, n2 + 1 , G × K2 admits C2z -covering for every z ∈ [2, m]. Let H = x1 x2 . . . xz be a subgraph of G isomorphic to Pz for an arbitrary z ∈ [2, m]. For each H, denote by H ′ = x′1 x′2 . . . x′z the corresponding subgraph in G′ . Thus, for each H, we obtain H ′′ as the corresponding subgraph in G × K2 isomorphic to C2z where V (H ′′ ) = V (H) ∪ V (H ′ ) and E(H ′′ ) = E(H) ∪ E(H ′ ) ∪ {x1 x′1 , xz x′z }. We can verify that there are exactly t subgraphs of G × K2 isomorphic to C2z , where t is the number of subgraphs in G isomorphic to Pz . Thus, X X X X wtf (H ′′ ) = f (v) + f (v) + f (uv) + f (uv) + f (x1 x′1 ) + f (xz x′z ) v∈V (H) = X v∈V (H ′ ) [f (v) + f (v ′ )] + v∈V (H) = X hX v∈V (H) uv∈E(H) X uv∈E(H ′ ) [f (uv) + f (u′ v ′ )] + 3p + 1 − g(x1 ) + 3p + 1 − g(xz ) uv∈E(H) i Wg(v) + X X uv∈E(H)  Y 1 (2q+a−g(u)−g(v)) + 6p + 2 − g(x1 ) − g(xz ) 2 = 7zp + 3zq − 3q + az − a + 2, which is independent of H ′′ . Therefore, G × K2 is (C2x , C2y )-sim-supermagic for all x, y ∈ [2, m]. An example of the labeling in Theorem 4.3 can be seen in Figure 4. Figure 4. A (C4 , C2m )-sim-supermagic labeling of P6 × K2 for m = 3 and 4 Note that the preceding theorem enlarges the classes of graphs known to be C2m -supermagic, as stated in Table 3. For instance, since every path Pn was shown to be (3, 2)-EAV [27], an immediate consequence of Theorem 4.3 is that the ladder Pn × K2 is (C4 , C2m )-sim-supermagic for every m ∈ [3, ⌊ n2 ⌋ + 1]. In [2], Bača and Barrientos described a connection between an α-labeling and an (a, 2)-EAV labeling of graphs. An injective mapping f : V (G) → [0, |E(G)|] is said to be graceful labeling if |f (x)−f (y)| are distinct for each xy ∈ E(G). A graceful labeling f is called an α-labeling if there exists an integer λ such that for each edge xy either f (x) ≤ λ < f (y) or f (y) ≤ λ < f (x) [24]. 60 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. A graph G that admits an α-labeling is said to be an α-graph. From the definition of α-labeling, it follows that an α-graph is necessarily bipartite. Let {A, B} be the natural bipartition of the vertex set of an α-graph. Bača and Barrientos [2] presented the following theorem. Theorem 4.4. [2] A tree T is a (3, 2)-EAV graph if and only if T is an α-graph and ||A|−|B|| ≤ 1, where {A, B} is the natural bipartition of the vertex set of T . Theorem 4.3 together with Theorem 4.4 implies the relationship between an α-labeling of a tree T and a (C4 , C6 )-sim-supermagic labeling of the Cartesian product T × K2 . Let n ≥ 2 be a positive integer and let T be an α-tree and ||A| − |B|| ≤ 1, where {A, B} is the natural bipartition of the vertex set of T . It is clear that T × K2 admits a C4 -covering and a C6 -covering only if T is not isomorphic to a star. Corollary 4.1. Let T be an α-tree not isomorphic to a star on at least five vertices and let ||A| − |B|| ≤ 1, where {A, B} is the natural bipartition of the vertex set of T . Then T × K2 is (C4 , C6 )sim-supermagic. Figure 5 illustrates a (C4 , C6 )-sim-supermagic labeling of product graph S2,1,0,1 × K2 . Figure 5. A (C4 , C6 )-sim-supermagic labeling of S2,1,0,1 × K2 . A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex of the graph is incident to exactly one edge of the matching. Brankovic et al. [8] posed the following conjecture for α-trees. Conjecture 1. [8] All trees with maximum degree three and a perfect matching have an α-labeling. Consider a tree T with a perfect matching. Since T is bipartite, by a perfect matching in T , we have a natural bipartition of the vertex-set of T , namely A and B, such that ||A| − |B|| ≤ 1. As a direct consequence of Corollary 4.1 and Conjecture 1, the following holds. Theorem 4.5. Let T be a tree on at least five vertices that are not isomorphic to a star, with a maximum degree three and containing a perfect matching. If Conjecture 1 is true, then T × K2 is (C4 , C6 )-sim-supermagic. 61 www.ejgta.org On (F, H)-simultaneously-magic labelings of graphs | Y.F. Ashari et al. Although all our results in this section are restricted to trees, the proof of Theorem 7 in [16] implied that Cn × K2 is (C4 , C2m )-sim-supermagic for each odd n ≥ 5 and m = 2. Thus, it is interesting to seek conditions such that a Cartesian product of a non-tree graph G with K2 admits a (C4 , C2m )-sim-supermagic labeling. Acknowledgement Y.F. Ashari is supported by the PKPI/Sandwich-like-PMDSU Scholarship funded by the Indonesian Ministry of Research and Technology. Salman is supported by ”Program Penelitian dan Pengabdian kepada Masyarakat-Institut Teknologi Bandung” (P3MI-ITB). R. Simanjuntak is supported by Penelitian Dasar Unggulan Perguruan Tinggi 2021-2023 No. 2/E1/KP.PTNBH/2021 funded by Indonesian Ministry of Education, Culture, Research and Technology. A. SemaničováFeňovčı́ková and M. 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