Iranian Journal of Mathematical Sciences and Informatics Vol. 17, No. 1 (2022), pp 153-163 DOI: 10.52547/ijmsi.17.1.153 On the Volume of µ-way G-trade Narjes Khatoun Khademian, Nasrin Soltankhah∗ Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran E-mail: ir.khademian@gmail.com E-mail: soltan@alzahra.ac.ir Abstract. A µ-way G-trade (µ ≥ 2) consists of µ disjoint decompositions of some simple (underlying) graph H into copies of a graph G. The number of copies of the graph G in each of the decompositions is the volume of the G-trade and denoted by s. In this paper, we determine all values s for which there exists a µ-way K1,m -trade of volume s for underlying graph H = K2m,2m and H = K2m . [ Downloaded from ijmsi.ir on 2022-05-09 ] Keywords: Trade, G-trade, µ-way G-trade ,Trade spectrum. 2000 Mathematics subject classification: 05B99. 1. Introduction Trades are useful combinatorial objects with many applications in various areas of combinatorial configurations. In other words, a combinatorial trade is a subset of a combinatorial configuration which may be “exchanged” without changing overall parameters in the configuration. The combinatorial configuration may be a block design, a latin square, or in the case of this literature, a graph. The concept of trade was first introduced in the block designs by Hedayat (see [16]), however trades were used back in 1917 by White, Cole and Cummings [12]. Many papers on trades in block designs are concentrated on the existence ∗ Corresponding Author [ DOI: 10.52547/ijmsi.17.1.153 ] Received 2 October 2019; Accepted 14 December 2020 c 2022 Academic Center for Education, Culture and Research TMU 153 154 N. K. Khademian, N. Soltankhah and non-existence of trades (see [4, 17, 21, 26]). Trades were used in the latin squares, named as latin trades (see [5, 18, 19]). Afterwords trades have been introduced in graph theory, named as graphical trades (see [8, 27]). A decomposition of a graph H is a collection of edge-disjoint subgraphs of H, such that partitions the edges of H. If each of the subgraphs is isomorphic to some graph G, then the decomposition is called a G-decomposition of H. Suppose G is a simple graph. A G-trade of volume s is a pair {T1 , T2 } where each Ti (i = 1, 2) consists of s pairwise edge-disjoint graphs isomorphic to G. The s copies in T1 are distinct from the s copies in T2 and the edge-set of the graphs in T1 and T2 are identical and forming a simple graph say underlying graph H. Therefore T1 and T2 are two disjoint G-decompositions of underlying graph H. The number of vertices in the underlying graph H is the foundation of G-trade and denoted by v. In some papers on trades in graph theory, the underlying graph is unrestricted and it is obtained from the union of blocks in each Ti (see [8, 9, 24, 25]). In some other works, the underlying graph is a fixed graph with definite vertices and edges (see [7, 10]). The concept of trades have been generalized in latin trades and block designs before, see [5, 28], also see([2, 14, 15, 29, 31]). Recently, the idea of generalization of G-trade is investigated in [20], which we explain as follows. Definition 1.1. A µ-way G-trade of volume s with underlying graph H consists of µ disjoint decompositions of graph H into s edge-disjoint copies of G. In other words: (1) (1) (2) (2) (µ) (µ) [ Downloaded from ijmsi.ir on 2022-05-09 ] T1 = {G1 , G2 , ..., G(1) s }, T2 = {G1 , G2 , ..., G(2) s }, .. . Tµ = {G1 , G2 , ..., G(µ) s }, Ti ∩ Tj = ∅ ∀1 ≤ i, j ≤ µ, Gji ≃ G ∀1 ≤ i ≤ s, 1 ≤ j ≤ µ, s [ i=1 s [ (1) Gi = (2) Gi = ... = (1) E(Gi ) (j) = s [ s [ (2) E(Gi ) = ... i=1 (l) 6= Gk (µ) Gi = H, i=1 i=1 i=1 Gi s [ 1 ≤ i, k ≤ s, s [ (µ) E(Gi ) = E(H), i=1 1 ≤ j, l ≤ µ, i 6= k, j 6= l. [ DOI: 10.52547/ijmsi.17.1.153 ] The number of vertices in the underlying graph H is the foundation of µ-way G-trade and denoted by v. Each Gi is called a block. On the Volume of µ-way G-trade 155 Moreover, this generalization is mentioned in [10] for a particular state when G is a cycle. As usual Kn denotes the complete graph on n vertices and Km,n , denotes the complete bipartite graph with parts of sizes m, n. A star is a complete bipartite graph K1,x that is called x-star. A K1,x with vertex set {a0 , b1 , b2 , ..., bx } and edge set {a0 bi | 1 ≤ i ≤ x} is denoted by [a0 : b1 , b2 , ..., bx ]. Example 1.2. Figure (1) is a 3-way K1,2 -trade of volume 2, with foundation 5 on underlying graph H1 = K1,4 , where V (H1 ) = {1, 2, 3, 4, 5}. In other words H1 = K1,4 is decomposed into three disjoint partition. Figure 1. 3-way K1.2 -trade of volume 2 [ DOI: 10.52547/ijmsi.17.1.153 ] [ Downloaded from ijmsi.ir on 2022-05-09 ] Example 1.3. Figure (2) is a 3-way K1,2 -trade of volume 3, with foundation 7 on underlying graph H2 = K1,6 , where V (H2 ) = {1, a, b, c, d, e, f }. In other words H2 = K1,6 is decomposed into three disjoint partition. Figure 2. 3-way K1,2 -trade of volume 3 Trades are also intimately connected with the so-called intersection problem for combinatorial structures. This basically asks, given two combinatorial structures with the same parameters, and based on the same underlying set, such as a pair of block designs, a pair of latin squares, or a pair of graphs, in how 156 N. K. Khademian, N. Soltankhah many ways may they intersect? So for two block designs, how many common blocks may there be? Let D1 and D2 be two block designs with the same parameters and the same set of points. Clearly if the blocks common to D1 and D2 are deleted, the remaining blocks T1 in D1 and T2 in D2 will form a trade T = {T1 , T2 }. So the possible volume T of the trade is intimately connected with the intersection problem for block designs. The intersection problem has also been considered for more than just pairs of combinatorial structures; the intersection of µ combinatorial structures with µ > 2 was dealt with in, for example, [1, 3, 30] for three block desgins and [6, 11, 13] for three latin squares. These correspond in the same manner to µ-way G- trades in the corresponding combinatorial structure. [ Downloaded from ijmsi.ir on 2022-05-09 ] This gives one of the motivations to study the spectrum of volumes of trades, besides that the problem of determining the possible size of combinatorial objects from some studied class is very natural, itself. So the problem of determining the conditions for the existence, non-existence and the spectrum of volumes of µ-way G- trades is an important problem in combinatorial subjects. Not much is known for the mentioned questions on µ-way G- trades for µ ≥ 3 and most of the literature focused mainly on the case µ = 2, see [22, 23]. Here, we determine all values s for which there exists a µ-way K1,m -trade of volume s for underlying graph H = K2m,2m and H = K2m . Definition 1.4. The trade spectrum T Sµ (G) of G, is the set of values s for which there exists a µ-way G-trade of volume s. Let Xµ (G) denote the set of non-negative integers s for which no µ -way G-trade of volume s exists. The trade spectrum of G is additive: if x, y ∈ T Sµ (G), then certainly x+y ∈ T Sµ (G), just take the union of two trades, of volume x and the other of volume y. So according to the above examples, we have a 3-way K1,2 -trade of volume 2 + 3 = 5 on underlying graph H = K1,10 , where V (H) = V (H1 ) ∪ V (H2 ) = {1, 2, 3, 4, 5, a, b, c, d, e, f }. Note that T Sµ (K2 ) = {0} because T Sµ (K2 ) ⊆ T S(K2 ) and T S(K2 ) = {0}, where T S(K2 ) is the set of values for which there exists a K2 -trade of volume s, for µ = 2. Obviously 0 ∈ T Sµ (G), moreover, Xµ (G) and T Sµ (G) partition the set of non-negative integers. Certainly 1 ∈ Xµ (G), unless G contains an isolated vertex( see Lemma (2.1)). If 1 ∈ T Sµ (G), then T Sµ (G) contains all the set of non-negative integers (because T Sµ (G) of G is additive). 2. Some general results [ DOI: 10.52547/ijmsi.17.1.153 ] We start this section with general results for all graphs. On the Volume of µ-way G-trade 157 lemma 2.1. There exists a µ-way G-trade of each volume (Xµ (G) = ∅) if and only if G has isolated vertices. Proof. Certainly Xµ (G) = ∅ if and only if 1 ∈ T Sµ (G) and µ different copies of G can have the same edge set if and only if G has isolated vertices.  Example 2.2. Suppose G is a graph with one edge and one isolated vertex. In Figure (3) , there exists a 4-way G-trade of volume 1, with foundation 3 on underlying graph H, where H is a graph with 1 edge and 3 vertices, and here H = G ≃ G(1) ≃ G(2) ≃ G(3) ≃ G(4) . Figure 3. 4-way G-trade of volume 1 Theorem 2.3. Let I be an independent subset of V (G), such that G \ I is not connected graph, then Xµ (G) ⊆ {1, 2, . . . , µ − 1} (for t ≥ µ, t ∈ Tµ (G)). [ DOI: 10.52547/ijmsi.17.1.153 ] [ Downloaded from ijmsi.ir on 2022-05-09 ] Proof. We may assume that G is non-empty. Let t ≥ µ, we need only show that t ∈ T Sµ (G). Let C be the vertex set of a component of G \ I, and D be the rest of the vertices of G \ I. Let Zt denote the ring of integers modulo t. S S For each i ∈ Zt place copy Gi of G on the vertex set C × {i} I D × {i} respectively and we define: S S T1 : {(C × {i}) I (D × {i}); i ∈ Zt } S S T2 : {(C × {i}) I (D × {i + 1}); i ∈ Zt } .. . S S Tµ : {(C × {i}) I (D × {i + µ − 1}) ; i ∈ Zt } Obviously T = {T1 , T2 , T3 . . . Tµ } is a µ-way G-trade of volume t.  Example 2.4. The trade in Figure (4) is constructed according to Theorem (2.3). Let G be a triangle graph with one hanging edge (G = K4 \ K1,2 ), with vertex set V (G) = {1, 2, 3, 4} and with edge set E(G) = {12, 23, 13, 34}. Take I = {3}, C = {1, 2} and D = {4}. It is clear that G \ I is not connected graph and 3 ∈ T3 (G). The following trade is a 3-way G-trade of volume 3. S S T1 : {Gi ; i ∈ Z3 } = {({1, 2} × {i}) 3 (4 × {i}); i ∈ Z3 } S S ′ T2 : {Gi ; i ∈ Z3 } = {({1, 2} × {i}) 3 (4 × {i + 1}); i ∈ Z3 } 158 N. K. Khademian, N. Soltankhah ′′ T3 : {Gi ; i ∈ Z3 } = {({1, 2} × {i}) S S 3 (4 × {i + 2}); i ∈ Z3 } The underlying graph H has 10 vertices and 12 edges as follows: V (H) ={(1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), 3, (4, 0), (4, 1), (4, 2)}, E(H) ={3(1, 0), 3(1, 1), 3(1, 2), 3(2, 0), 3(2, 1), 3(2, 2), (1, 0)(2, 0), (1, 1)(2, 1), (1, 2)(2, 2), 3(4, 0), 3(4, 1), 3(4, 2)} Figure 4. 3-way G-trade of volume 3 [ DOI: 10.52547/ijmsi.17.1.153 ] [ Downloaded from ijmsi.ir on 2022-05-09 ] Theorem 2.5. If for some independent subset I ⊆ V (G), the graph G \ I has k components C1 , C2 , . . . , Ck , (k ≥ µ), then Xµ (G) ⊆ {1}. Proof. Let t ≥ 2, we need only show that t ∈ T Sµ (G). Let Zt denote the ring of integers modulo t, then we define: S S S S T1 : {C1 × {i} C2 × {i} . . . Ck × {i} I ; i ∈ Zt } S S S S T2 : {C1 × {i} C2 × {i + 1} . . . Ck × {i} I ; i ∈ Zt } .. . S S S S Tk : {C1 × {i} C2 × {i} · · · Ck × {i + 1} I ; i ∈ Zt } Obviously T = {T1 , T2 , T3 . . . Tk } is a k-way G-trade of volume t and k ≥ µ, so there is a µ-way G-trade of volume t.  Example 2.6. The trade in Figure (5) is constructed according to Theorem (2.5). Let G be a tree graph with vertex set V (G) = {1, 2, 3, 4, 5} and with edge set E(G) = {12, 23, 34, 35}. Take I = {3}, C1 = {1, 2}, C2 = {4} and C3 = {5}. It is clear that G \ I is not connected and 2 ∈ T3 (G). The following trade is a On the Volume of µ-way G-trade 159 3-way G-trade of volume 2. [[ [ {5} × {i} {3} ; i ∈ Z2 } [ [[ [ ′ T2 :{Gi ; i ∈ Z2 } = {{1, 2} × {i} {4} × {i + 1} {5} × {i} {3} ; i ∈ Z2 } [ [[ [ ′′ T3 :{Gi ; i ∈ Z2 } = {{1, 2} × {i} {4} × {i} {5} × {i + 1} {3} ; i ∈ Z2 } T1 :{Gi ; i ∈ Z2 } = {{1, 2} × {i} [ {4} × {i} The underlying graph H has 9 vertices and 8 edges as follows: V (H) ={(1, 0), (1, 1), (2, 0), (2, 1), 3, (4, 0), (4, 1), (5, 0), (5, 1)}, [ DOI: 10.52547/ijmsi.17.1.153 ] [ Downloaded from ijmsi.ir on 2022-05-09 ] E(H) ={(1, 0)(2, 0), (1, 1)(2, 1), 3(2, 0), 3(2, 1), 3(4, 0), 3(4, 1), 3(5, 0), 3(5, 1)} Figure 5. 3-way G-trade of volume 2 3. µ-way K1,m -trade for H = K2m,2m In this section we determine the µ-way K1,m -trade spectrum for underlying graph H = K2m,2m . lemma 3.1. Let m and s be integers. If there  exists  a µ-way K1,m -trade of 2m m volume s on H = K1,2m , then s = 2 and µ ≤ . 2 160 Proof. There exist N. K. Khademian, N. Soltankhah  2m m  graphs K1,m in K1,2m . Since each Ti has s blocks,   2m 2m m then m × s = 2m, therefore s = = 2 and µ ≤ .  m 2   2m m , then there exists a µ-way Theorem 3.2. Let H = K1,2m and µ ≤ 2 K1,m -trade of volume 2. Proof. Let H = [0 : 1, 2, ..., 2m]. Without loss of generality let the edge 01 is contained in the firstblock ofeach Ti . The number of K1,m which include 2m − 1 the edge 01 is equal to . The second block in each Ti is determined m−1   2m     1 2m 2m − 1 m uniquely. It is clear = , then µ ≤ .  2 m m−1 2 [ Downloaded from ijmsi.ir on 2022-05-09 ] lemma 3.3. Let m and s be integers. If there exists  a µ-way K1,m -trade of 2m volume s on H = K2m,2m , then s = 4m and µ ≤ . m Proof. Since µ-way K1,m -trade is of volume s, then m × s = 4m2 , therefore 4m2 = 4m. Let V (H) = A ∪ B, where A and B are two disjoint sets and s= m |A| = |B| = 2m, and all edges are between A and B. There exist 2m graphs K1,2m with centres in A and 2m, K1,2m withcentres  in B. By Lemma (3.1) each 2m       4m × 2m 2m 2m m K1,2m has , K1,m . So we have and µ ≤ . = m 4m m m    2m Theorem 3.4. Let H = K2m,2m and µ ≤ , then there exists a µ-way m K1,m -trade of volume 4m. Proof. The result follows from Theorem (3.2).  4. µ-way K1,m -trade for H = K2m In this section we determine the µ-way K1,m trade spectrum on underlying graph H = K2m . [ DOI: 10.52547/ijmsi.17.1.153 ] lemma 4.1. Let m and s be integers. If there exists a µ-way K1,m   − trade of 2m − 1 2m × m volume s on H = K2m , then s = 2m − 1 and µ ≤ . 2m − 1 On the Volume of µ-way G-trade 161     2m − 1 degvi = 2m × graphs K1,m in K2m . m m vi ∈v(H)   2m Since µ-way K1,m -trade is of volume s, then s × m = , therefore s = 2     2m 2m − 1 2m × 2 m = 2m − 1 and µ ≤ . m 2m − 1  Proof. There exist P Billington and Hoffman in [7] obtained a 2-way K1,m -trade of volume 2m−1 for underlying graph H = K2m . Now we introduce two other disjoint K1,m decomposition for K2m . Theorem 4.2. Let H = K2m (m > 2). Then there exists a 4-way K1,m − trade of volume s = 2m − 1. Proof. Let V (H) = {0, 1, 2, ...2m − 1} be vertex set of underlying graph H. ′ Suppose K1,m in T1 , T2 be [i : j1 , j2 , ..., jm ], [i : j1′ , j2′ , ..., jm ] for i, i′ , jk , jk′ ∈ 0, 1, ..., 2m − 1, as constructed in [7]. Now we construct T3 , T4 as follows: T3 : [i + m : j1 + m, j2 + m, ..., jm + m] [ DOI: 10.52547/ijmsi.17.1.153 ] [ Downloaded from ijmsi.ir on 2022-05-09 ] ′ T4 : [i + m : j1′ + m, j2′ + m, ..., jm + m] with modulo 2m. We show that T2 and T3 are disjoint. Otherwise, assume that Bi′ = Bj′′ = [α + m : β1 + m, β2 + m, ..., βm + m], where Bi′ ∈ T2 and Bj′′ ∈ T3 . Therefore there exists a Bk ∈ T1 that Bk = [α : β1 , β2 , ..., βm ]. There exists a βi + m = α + m − 1. Hence the BK in T1 is following: BK = [α : ..., α − 1, ..., α + 1, ...]. Therefore exists a block in T2 that is [α : ..., α − 1, ...α − 1, ...]. This gives a contradiction.  Example 4.3. 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