On imbalances in oriented multipartite graphs

Transcription

1 Acta Univ. Sapientiae, Mathematica, 3, 1 (2011) On imbalances in oriented multipartite graphs S. Pirzada King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia sdpirzada@yahoo.co.in Abdulaziz M. Al-Assaf King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia alassaf@fupm@edu.sa Koo K. Kayibi King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia ayibi@fupm.edu.sa Abstract. An oriented -partite graph(multipartite graph) is the result of assigning a direction to each edge of a simple -partite graph. Let D(V 1,V 2,,V ) be an oriented -partite graph, and let d + v ij and d v ij be respectively the outdegree and indegree of a vertex v ij in V i. Define b vij (or simply b ij as b ij = d + v ij d v ij as the imbalance of the vertex v ij. In this paper, we characterize the imbalances of oriented -partite graphs and give a constructive and existence criteria for sequences of integers to be the imbalances of some oriented -partite graph. Also, we show the existence of an oriented -partite graph with the given imbalance set. 1 Introduction A digraph without loops and without multi-arcs is called a simple digraph. Mubayi et al. [1] defined the imbalance of a vertex v i in a digraph as b vi (or simply b i ) = d + v i d v i, where d + v i and d v i are respectively the outdegree and indegree of v i. The imbalance sequence of a simple digraph is formed by 2010 Mathematics Subject Classification: 05C20 Key words and phrases: digaph, imbalance, outdegree, indegree, oriented graph, oriented multipartite graph, arc 34

2 On imbalances in oriented graphs 35 listing the vertex imbalances in non-increasing order. A sequence of integers F = [f 1, f 2,, f n ] with f 1 f 2 f n is feasible if it has sum zero and satisfies f i (n ), for 1 < n. The following result [1] provides a necessary and sufficient condition for a sequence of integers to be the imbalance sequence of a simple digraph. Theorem 1 A sequence is realizable as an imbalance sequence if and only if it is feasible. The above result is equivalent to saying that a sequence of integers B = [b 1, b 2,, b n ] with b 1 b 2 b n is an imbalance sequence of a simple digraph if and only if for 1 < n b i (n ), with equality when = n. On arranging the imbalance sequence in non-decreasing order, we have the following observation. Theorem 2 A sequence of integers B = [b 1, b 2,, b n ] with b 1 b 2 b n is an imbalance sequence of a simple digraph if and only if for 1 < n with equality when = n. b i (n ), Various results for imbalances in digraphs and oriented graphs can be found in [2, 3, 4, 5]. 2 Imbalance sequences in oriented multipartite graphs An oriented multipartite (-partite) graph is the result of assigning a direction to each edge of a simple multipartite (-partite) graph, 2. Throughout this paper we denote an oriented -partite graph by -OG, unless otherwise stated. Let V i = {v i1, v i2,, v ini }, 1 i, be parts of -OG D(V 1, V 2,, V ),

3 36 S. Pirzada, A. M. Al-Assaf, K. K. Kayibi and let d + v ij and d v ij, 1 j n i, be respectively the outdegree and indegree of a vertex v ij in V i. Define b vij (or simply b ij as b ij = d + v ij d v ij as the imbalance of the vertex v ij. The sequences B i = [b i1, b i2,, b ini ], 1 i, in nondecreasing order are called the imbalance sequences of D(V 1, V 2,, V ). The sequences of integers B i = [b i1, b i2,, b ini ], 1 i, in nondecreasing order are said to be realizable if there exists an -OG with imbalance sequences B i, 1 i. Various criterions for imbalance sequences in -OG can be found in [2]. For any two vertices v ij in V i and v lm in V l (i l, 1 i l, 1 j n i, 1 m n l ) of -OG D(V 1, V 2,, V ), we have one of the following possibilities. (i). An arc directed from v ij to v lm, denoted by v ij (1 0)v lm. (ii). An arc directed from v lm to v ij, denoted by v ij (0 1)v lm. (iii). There is no arc from v ij to v lm and there is no arc from v lm to v ij and this is denoted by v ij (0 0)v lm. A triple in -OG is an induced suboriented graph of three vertices with exactly one vertex from each part. For any three vertices v ij, v lm and v pq in -OG D, the triples of the form v ij (1 0)v lm (1 0)v pq (1 0)v ij, or v ij (1 0)v lm (1 0)v pq (0 0)v ij are said to be oriented intransitive, while as the triples of the form v ij (1 0)v lm (1 0)v pq (0 1)v ij, or v ij (1 0)v lm (0 1)v pq (0 0)v ij, or v ij (1 0)v lm (0 0)v pq (0 1)v ij, or v ij (1 0)v lm (0 0)v pq (0 0)v ij, or v ij (0 0)v lm (0 0)v pq (0 0)v ij are said to be oriented transitive. An -OG is said to be oriented transitive if all its triples are oriented transitive, otherwise oriented intransitive. We have the following observation. Theorem 3 Let D and D be two -OG with the same imbalance sequences. Then D can be transformed to D by successively transforming appropriate triples in one of the following ways. Either (a) by changing a cyclic triple v ij (1 0)v lm (1 0)v pq (1 0)v ij to an oriented transitive triple v ij (0 0)v lm (0 0)v pq (0 0)v ij which has the same imbalance sequences, or vice versa, or (b) by changing an oriented intransitive triple v ij (1 0)v lm (1 0)v pq (0 0)v ij to an oriented transitive triple v ij (0 0)v lm (0 0)v pq (0 1)v ij which has the same imbalance sequences, or vice versa. Proof. Let B i be the imbalance sequences of -OG D whose parts are V i, 1 i and V i = n i. Let D be -OG with parts V i, 1 i. To prove the result, it is sufficent to show that D can be obtained from D by successively transforming triples in any one of the ways as given in (a), or (b).

4 On imbalances in oriented graphs 37 We fix n i, 2 i and use induction on n 1. For n 1 = 1, the result is obvious. Assume that the result holds when there are fewer than n 1 vertices in the first part. Let j 2, j 3,, j be such that for l 2 > j 2, l 3 > j 3,, l > j, 1 j 2 < l 2 n 2, 1 j 3 < l 3 n 3,, 1 j < l n, the corresponding arcs have the same orientations in D and D. For j 2, j 3,, j and 2 i, p, q, p q, we have three cases to consider. (i). v 1n1 (1 0)v ijp (1 0)v ijq and v 1n 1 (0 0)v ij p, (ii). v 1n1 (0 0)v ijp (0 1)v ijq and v 1n 1 (1 0)v ij p (0 0)v ij q and (iii). v 1n1 (1 0)v ijp (0 0)v ijq and v 1n 1 (0 0)v ij p (0 1)v ij q. Case (i). Since v 1n1 and v 1n 1 have equal imbalances, we have v 1n1 (0 1)v ijq and v 1n 1 (0 0)v ij q, or v 1n1 (0 0)v ijq and v 1n 1 (1 0)v ij q. Thus there is a triple v 1n1 (1 0)v ijp (1 0)v ijq (1 0)v 1n1, or v 1n1 (1 0)v ijp (1 0)v ijq (0 0)v 1n1 in D, and corresponding to these v 1n 1 (0 0)v ij p (0 0)v 1n 1, or v 1n 1 (0 0)v ij p (0 1)v 1n 1 respectively is a triple in D. Case (ii). Since v 1n1 and v 1n 1 have equal imbalances, we have v 1n1 (1 0)v ijq and v 1n 1. Thus there is a triple v 1n1 (0 0)v ijp (0 1)v ijq (0 1)v 1n1 in D and corresponding to this v 1n 1 (1 0)v ij p (0 0)v 1n 1 is a triple in D. Case (iii). Since v 1n1 and v 1n 1 have equal imbalances, therefore we have v 1n1 (0 1)v ijq and v 1n 1. Thus v 1n1 (1 0)v ijp (0 0)v ijq (1 0)v 1n1 is a triple in D, and corresponding to this v 1n 1 (0 0)v ij p (0 1)v ij q (0 0)v 1n 1 is a triple in D. Therefore from (i), (ii) and (iii) it follows that there is an -OG that can be obtained from D by any one of the transformations (a) or (b) with the imbalances remaining unchanged. Hence the result follows by induction. Corollary 1 Among all -OG with given imbalance sequences, those with the fewest arcs are oriented transitive. A transmitter is a vertex with indegree zero. In a transitive oriented -OG with imbalance sequences B i = [b i1, b i2,, b ini ], 1 i, any of the vertices with imbalances b ini, can act as a transmitter. The next result provides a useful recursive test of checing whether the sequences of integers are the imbalance sequences of -OG. Theorem 4 Let B i = [b i1, b i2,, b ini ], 1 i, be sequences of integers in non-decreasing order with b 1n1 > 0 and b jnj r=1,r j n r, for all j, 2 i. Let B 1 be obtained from B 1 by deleting one entry b 1n1, and let

5 38 S. Pirzada, A. M. Al-Assaf, K. K. Kayibi B 2, B 3,, B, be obtained from B 2, B 3,, B by increasing b 1n1 smallest entries of B 2, B 3,, B by one each. Then B i are imbalance sequences of some -OG if and only if B i are imbalance sequences. Proof. Suppose B i be the imbalance sequences of some -OG D with parts V i, 1 i. Then -OG D with imbalance sequences B i can be obtained by adding a vertex v 1n1 in V 1 such that v 1n 1 (1 0)v ij for those vertices v ij in V i, i 1 whose imbalances are increased by one in going from B i to B i. Conversely, let B i be the imbalance sequences of -OG D with parts V i, 1 i. By Corollary 4, any of the vertices v ini in V i with imbalances b ini, 1 i can be a transmitter. Assume that the vertex v 1n1 in V 1 with imbalance b 1n1 be a transmitter. Clearly, d + v 1n1 > 0 and d v 1n1 = 0 so that b 1n1 = d + v 1n1 d v 1n1 > 0. Also, d + v jnj r=1,r j n r and d v jnj 0 for 2 i so that b jnj = d + v jnj d v jnj r=1,r j n r. Let U be the set of v 1n1 vertices of smallest imbalances in V j, 2 i and let W = V 2 V 3 V U. Now construct D such that v 1n1 (1 0)u for all u in U. Clearly D {v 1n1 } realizes V i, 1 i. Theorem 5 provides an algorithm for determining whether or not the sequences B i, 1 i of integers in non-decreasing order are the imbalance sequences and for constructing a corresponding -OG. Suppose B i = [b i1, b i2,, b ini ], 1 i, be imbalance sequences of -OG with parts V i = {v i1, v i2,, v ini }, where b 1n1 > 0 and b jnj r=1,r j n r, 2 i. Deleting b 1n1 and increasing b 1n1 smallest entries of B 2, B 3,, B by 1 each to form B 2, B 3,, B. Then arcs are defined by v 1n 1 (1 0)v ij, for which b v ij = b vij + 1, where i 1. If at least one of the conditions b 1n1 > 0, or b jnj r=1,r j n r does not hold, then we delete b ini for that i for which the conditions get satisfied and the same argument is used for defining arcs. If this method is applied recursively, then (i) it tests whether B i are the imbalance sequences, and if B i are the imbalance sequences (ii) -OG (B i ) with imbalance sequences B i is constructed. We illustrate this reduction and resulting construction as follows. Consider the four sequences B 1 = [1, 3, 4], B 2 = [ 3, 2, 2], B 3 = [ 4, 3] and B 4 = [ 3, 1]. (i) [1, 3, 4], [ 3, 2, 2], [ 4, 3], [ 3, 1] (ii) [1, 3], [ 2, 2, 2], [ 3, 2], [ 2, 1], v 13 (1 0)v 21, v 13 (1 0)v 31, v 13 (1 0)v 32, v 13 (1 0)v 41 (iii) [1], [ 1, 2, 2], [ 2, 1], [ 2, 1], v 12 (1 0)v 21, v 12 (1 0)v 31, v 12 (1 0)v 32 (iv), [ 1, 2, 2], [ 2, 1], [ 2, 1], v 11 (1 0)v 31

6 On imbalances in oriented graphs 39 (v), [ 1, 2], [0, 1], [ 1, 1], v 23 (1 0)v 31, v 23 (1 0)v 41 or,, [ 1, 2], [ 1, 0], [ 1, 1] (vi), [ 1], [0, 0], [0, 1], v 22 (1 0)v 32, v 22 (1 0)v 41 (vii), [0], [0, 0], [0, 0], v 42 (1 0)v 21. Clearly 4-OG with parts V 1 = {v 11, v 12, v 13 }, V 2 = {v 21, v 22, v 23 }, V 3 = {v 31, v 32 } and V 4 = {v 41, v 42 } in which v 13 (1 0)v 21, v 13 (1 0)v 31, v 13 (1 0)v 32, v 13 (1 0)v 41, v 12 (1 0)v 21, v 12 (1 0)v 31, v 12 (1 0)v 32, v 11 (1 0)v 31, v 23 (1 0)v 31, v 23 (1 0)v 41, v 22 (1 0)v 32, v 22 (1 0)v 41, v 42 (1 0)v 21 are arcs has imbalance sequences [1, 3, 4], [ 3, 2, 2], [ 4, 3] and [ 3, 1]. The next result gives a combinatorial criterion for determining whether sequences of integers are realizable as imbalances. Theorem 5 Let B i = [b i1, b i2,, b ini ], 1 i, be sequences of integers in non-decreasing order. Then B i are the imbalance sequences of some -OG if and only if m i 1 b ij 2 j=i+1 m i m j n i m j m i n i, (1) for all sets of integers m i, 0 m i n i with equality when m i = n i. Proof. The necessity of the condition follows from the fact that the -OG induced by m i vertices for 1 i, 1 m i n i has a sum of imbalances 2 1 j=i+1 m im j n i m j m in i. For sufficiency, assume that B i = [b i1, b i2,, b ini ], 1 i be the sequences of integers in non-decreasing order satisfying conditions (1) but are not the imbalance sequences of any -OG. Let these sequences be chosen in such a way that n i, 1 i are the smallest possible and b 11 is the least for the choice of n i. We consider the following two cases. Case (i). Suppose equality in (1) holds for some m j n j, 1 i 1, m n, so that m i 1 b ij = 2 j=i+1 m i m j n i m j m i n i. By the minimality of n i, 1 i the sequences B i = [b i1, b i2,, b imi ] are the imbalance sequences of some -OG D (V 1, V 2,, V ). Define B i = [b i(m i +1), b i(mi +2),, b i(ni )], 1 i.

7 40 S. Pirzada, A. M. Al-Assaf, K. K. Kayibi Consider the sum f i b i(mi +j) = 1 2 m i +f i j=i+1 b ij m i b ij (m i + f i )(m j + f j ) 1 (m i + f i )n i 2 1 = 2 + j=i+1 m i n i j=i+1 1 m i m j m i n i 2 j=i+1 1 f i n i 2 j=i+1 n i (m j + f j ) m i m j + n i m j + (m i f j + f i m j + f i f j ) j=i+1 f i f j m i m j + m i n i n i n i m j + n i f j f i n i, for 1 f i n i m i, with equality when f i = n i m i for all i, 1 i. So by the minimality of n i, 1 i, the sequences B i form the imbalance sequence of some -OG D (V 1, V 2,, V ). Construct a new -OG D(V 1, V 2,, V ) as follows. Let V 1 = V 1 V 1, V 2 = V 2 V 2, V = V V with V i V i = and the arc set containing those arcs which are among V 1, V 2,, V and among V 1, V 2,, V. Then D(V 1, V 2,, V ) has imbalance sequences B i, 1 i, which is a contradiction. Case (ii). Assume that the strict inequality holds in (1) for some m i n i, 1 i. Let B 1 = [b 11 1, b 12,, b 1n1 1, b 1n1 ] and let B j = [b j1, b j2,, b jnj ] for all j, 2 j. Clearly the sequences B i, 1 i satisfy conditions (1). Therefore, by the minimality of b 11, the sequences B i, 1 i are the imbalance sequences of some -OG D (V 1, V 2,, V ). Let b v 11 = b 11 1 and b v1n1 = b 1n Since b v1n1 > b v11 + 1, there exists a vertex v ij either in V i, 1 i, 1 j n i, such that v 1n1 (0 0)v ij (1 0)v 11, or v 1n1 (1 0)v ij (0 0)v 11, or v 1n1 (1 0)v ij (1 0)v 11, or v 1n1 (0 0)v ij (0 0)v 11 in D (V 1, V 2,, V ), and if these are changed to v 1n1 (0 1)v ij (0 0)v 11, or v 1n1 (0 0)v ij (0 1)v 11, or v 1n1 (0 0)v ij (0 0)v 11, or v 1n1 (0 1)v ij (0 1)v 11 respectively, the result is -OG with imbalance sequences B i, which is a contradiction. This completes the proof. f j

8 On imbalances in oriented graphs 41 3 Imbalance sets in oriented multipartite graphs The set of distinct imbalances of the vertices in -OG is called its imbalance set. Now we give the existence of -OG with a given imbalance set. Theorem 6 Let S = {s 1, s 2,, s n } and T = { t 1, t 2,, t n }, where s 1, s 2,, s n, t 1, t 2,, t n are positive integers with s 1 < s 2 < < s n and t 1 < t 2 < < t n. Then there exists -OG with imbalance set S T. Proof. First assume that 2 is even. Construct -OG D(V 1, V 2,, V ) as follows. Let V 1 = V 11 V 12 V 1n, V 2 = V 21 V 22 V 2n,... V = V 1 V 2 V n, with V ij V lm =, V ij = t i for all odd i, 1 i 1, 1 j n and V ij = s i for all even i, 2 i, 1 j n. Let there be an arc from each vertex of V ij to every vertex of V (i+1) j for all odd i, 1 i 1, 1 j n so that we obtain -OG with imbalance of vertices as follows. For odd i, 1 i 1 and 1 j n b vij = V (i+1)j 0 = s i, for all v ij V ij ; and for even i, 2 i and 1 j n b vij = 0 V (i+1)j = t i, for all v ij V ij Therefore imbalance set of D(V 1, V 2,, V ) is S T. Now assume 3 is odd. Construct -OG D(V 1, V 2,, V ) as below. Let V 1 = V 11 V 11 V 12 V V 1n V 1n, V 2 = V 21 V V 2n,... V 1 = V ( 1)1 V ( 1)2... V ( 1)n, V = V 1 V 2... V n,

9 42 S. Pirzada, A. M. Al-Assaf, K. K. Kayibi with V ij V lm =, V ij V lm =, V ij V lm =, V ij = t i for all i, 1 i 2, 1 j n, V i j = s i for all even i, 2 i 1, 1 j n, V ij = t i for all j, 1 j n and V j = s j for all j, 1 j n. Let there be an arc from each vertex of V ij to every vertex of V (i+1)j for all i, 1 i 2, 1 j n and let there be an arc from each vertex of V 1j to every vertex of V j for all j, 1 j n, so that we obtain -OG with imbalance set S T, as above. References [1] D. Mubayi, T. G. Will, D. B. West, Realizing degree imbalances in directed graphs, Discrete Math., 239 (2001), [2] S. Pirzada, On imbalances in digraphs, Kragujevac J. Math., 31 (2008), [3] S. Pirzada, T. A. Naioo, U. Samee, A. Iványi, Imbalances in multidigraphs, Acta Univ. Sapientiae, Math., 2 (2010), [4] S. Pirzada, T. A. Naioo, N. A. Shah, Imbalances in oriented tripartite graphs, Acta Math. Sinica, 27 (2011), [5] U. Samee, T. A. Cahisti, On imbalances in oriented bipartite graphs, Eurasian Math. J., 1 (2010), Received: October 20, 2010