Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs

Ain Shams Engineering Journal (2015) 6, 657–660

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs S. El-Serafi, R. El-Shanawany, H. Shabana

*

Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufiya University, Menouf, Egypt Received 19 January 2014; revised 15 November 2014; accepted 2 December 2014 Available online 29 January 2015

KEYWORDS Graph decomposition; Orthogonal double cover; Symmetric starter

Abstract Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with symmetric starter vectors of the orthogonal double covers (ODCs) of the complete bipartite graph and using the method of cartesian product of symmetric starter vectors to construct ODC of the complete bipartite graph by G, where G is a complete bipartite graph, disjoint union of different complete bipartite graphs and disjoint union of finite copies of a complete bipartite graph.  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction All graphs here are, finite, simple and undirected. Let H be any regular graph and let G ¼ fG0 ; G1 ; . . . ; GjVðHÞj1 g be a collection of ŒV(H)Œ subgraphs (pages) of H. G is an orthogonal double cover (ODC) of H if it has the following properties. i. Double cover property.Every edge of H is contained in exactly two of the pages in G.

* Corresponding author. Tel.: +20 483653541, +20 1210307205. E-mail addresses: Said_elserafi@yahoo.com (S. El-Serafi), [email protected] (R. El-Shanawany), [email protected] (H. Shabana). Peer review under responsibility of Ain Shams University.

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ii. Orthogonality propertyFor any two distinct pages Gi and Gj in GjEðGi Þ \ EðGj Þj ¼ 1 if and only if i and j are adjacent in H. If all pages Gi @ G for all i 2 {0, 1, . . . , Œ V(H)Œ  1}, then G is an ODC of H by G. For the definition of an orthogonal double cover (ODC) of the complete graph Kn by a graph G and for a survey on this topic, see [1]. While in principle any regular graph is worth considering. The choice of H = Kn,n is quite natural, also in view of a technical motivation: ODCs of such graphs are a helpful tool for constructing ODCs of Kn (see [4]). In this paper, we assume H = Kn,n the complete bipartite graph with partition sets of size n each. Furthermore we make use of the usual notation: D [ F for the disjoint union of D and F and mD for m disjoint copies of D. Denote the vertices of the partition sets of Kn,n by {00, 10, . . . , (n  1)0} and {01, 11, . . . , (n  1)1}. The length of an edge {x0, y1} of Kn,n is defined to be the difference y-x, where xy 2 Zn ¼ f0; 1; 2; . . . ; n  1g. Note that sums and differences

http://dx.doi.org/10.1016/j.asej.2014.12.002 2090-4479  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

658 are calculated in Zn (that is, sums and differences are calculated modulo n). If there is no danger of ambiguity if {x0, y1} 2 E(Kn,n) we can write {x0, y1} as x0y1. Let G be a subgraph of Kn,n and a 2 Zn . The a-translate of G, denoted by G + a is the edge-induced subgraph of Kn,n induced by {(x + a)0(y + a)1:x0y1 2 E(G)}. A subgraph G of Kn,n is called half-starter if ŒE(G)Œ = n and the lengths of all edges in G are mutually different. We denote a half-starter G by the vector v(G) = (v0, v1, . . . , vn1), where v0 ; v1 ; . . . ; vn1 2 Zn and vi can be obtained from the unique edge (vi)0(vi + i)1 of length i in G. Two half-starters v(G) = (v0, v1, . . . , vn1) and v(F) = (u0, u1, . . . , un1) are said to be orthogonal if fvi  ui : i 2 Zn g ¼ Zn . For a subgraph G of Kn,n with n edges, the edge-induced subgraph Gs with E(Gs) = {y0x1:x0y1 2 E(G)} is called the symmetric graph of G. El-Shanawany et al. [4] established following three theorems. Theorem 1.1. If G is a half-starter, then the collection of all translates of G forms an edge-decomposition of Kn,n. Theorem 1.2. If two half-starter v(G) and v(F) are orthogonal, then the union of the set of translates of G and the set of translates of F forms an ODC of Kn,n. Theorem 1.3. A half-starter is a symmetric starter if and only iff fvi  vi þ i : i 2 Zn g ¼ Zn : An algebraic construction of ODCs via symmetric starters has been exploited to get a complete classification of ODCs of Kn,n by G for n 6 9: a few exceptions apart, all graphs G are found this way (see [4]). Much of research on this subject focused with the detection of ODCs with pages isomorphic to a given graph G. For more results on ODCs, see [2,7]. In [3], the other terminologies not defined here can be found. In constructing ODCs a natural approach is to try to put two given ODCs somehow together to obtain ODCs of a larger complete bipartite graph. The following theorem of El-Shanawany et al. [6] relates the ODCs of complete bipartite graphs and the Cartesian product of two symmetric starter vectors. Theorem 1.4. [6] The Cartesian product of any two symmetric starter vectors is a symmetric starter vector with respect to the Cartesian product of the corresponding groups. It should be noted that v(D) · v(F) is not the usual cartesian product of the graphs D and F that has been studied widely in the literature. Our results based on the following symmetric starter vectors for a few classes of graphs that can be used as ingredients for cartesian product construction to obtain new symmetric starter vectors of an ODC of the complete bipartite graph by disjoint union of complete bipartite graphs. I. nK2 is a symmetric starter vector of an ODC of Kn,n and v(nK2) = (0, 1, 2, . . . , n  1) where n ” 1,5 mod 6, see [4]. II. K1,n is a symmetric starter vector of an ODC of Kn,n and v(K1,n) = (c, c, . . . , c) where 2 Zn , see [4]. III. Kr,s is a symmetric starter vector of an ODC of Kn,n and

S. El-Serafi et al. 0 s times 1 s times s times zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ vðKr;s Þ ¼ @0; 0; .. .; 0; sðr  1Þ; sðr  1Þ; .. . ;sðr  1Þ;   ;s; s; .. .; sA

where n = rs, See [5]. IV. 2K1,n is a symmetric starter vector of an ODC of K2n,2n and v(2K1,n) = (n, n  2, n, n  2, . . . , n, n  2) where n P 2 see [5]. V. K1,4 [ K2,1 [ K1,1 [ K1,2n-6 is a symmetric starter vector of an ODC of K2n+1, 2n+1 and v(K1,4 [ K2,1 [ K1,1 [ K1, 2n6) = (3, 2n, 2n, 2n  3, 0, 0, 0, . . . , 0, 0, 0, 2, 2n, 2n) where n P 4 see [5].

2. ODCs of Kmn,mn by complete bipartite graphs In the following, if there is no danger of ambiguity, we can write (a, b) as ab, if ða; bÞ 2 Zm  Zn (the Cartesian product of Zm and Zn Þ. Furthermore, we make the use of A for a subset of Zm , where, A = {x:x = s(r  j) for all 1 6 j 6 r} for a positive integer m such that m = rs. Theorem 2.1. Let nrs and mi be positive integers such that mi = risi for all 1 6 i 6 n. Then there exists an ODC of KQn mi ; Qn mi by KQn ri ; Qn si . i¼1

i¼1

i¼1

i¼1

Proof. According to the symmetric starter vector III, vðKri ;si Þ are symmetric starter vectors with respect to Zmi . Applying Theorem (1.4) proves 0 1 n times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ A n n vðKr1 ;s1 Þ  vðKr2 ;s2 Þ      vðKrn ;sn Þ ¼ v@KQ Q ri ;

i¼1

si

i¼1

Q is a symmetric starter vector with respect to ni¼1 Zmi . Moreover, the resulting symmetric starter graph has the following edges set: 0

1

B C B C EBKY n n C¼ Y @ A ri ;

i¼1

si

( u0 v1 : 8u 2

n Y

Ai ; v 2

n Y

i¼1

) Zsi

i¼1

i¼1

with Ai = {xi:xi = si(ri  j), 1 6 j 6 ri}.

h

Example 2.1. Let m1 = 6 and m2 = 4, according to the symmetric starter vector III, (1) (0, 0, 0, 3, 3, 3) is a symmetric starter vector of an ODC of K6,6 by K2,3 with respect to Z6 and (2) (0, 0, 2, 2) is a symmetric starter vector of an ODC of K4,4 by K2,2 with respect to Z4 . By Theorem (1.4), v(K2,3) · v(K2,2) = (00, 00, 02, 02, 00, 00, 02, 02, 00, 00, 02, 02, 30, 30, 32, 32, 30, 30, 32, 32, 30, 30, 32, 32) = v(K4,6) is a symmetric starter vector of an ODC of K24,24 with respect to Z6  Z4 . Furthermore, edges set of K4,6 can be constructed as follows, w.l.o.g. m1 = r1s1 = 6 ) r1 = 2, s1 = 3 and hence, A1 = {3(2  1), 3(2  2)} = {3, 0}. Similar-

Orthogonal double cover of Complete Bipartite Graph (00)0

(02)0

(00)1

Figure 1

(01)1

659 (30)0

(10)1

(32)0

(11)1

(20)1

(21)1

Symmetric starter of an ODC of K24,24 by K4,6 with respect to Z6  Z4 .

ity, m2 = r2s2 = 4 ) r2 = s2 = 2 and hence, A2 = {2(2  1), 2(2  2)} = {2, 0} then, EðKr1 r2 ;s1 s2 Þ ¼ EðK4;6 Þ ¼ fu0 v1 : v2 8u 2 A1  A2 ; v 2 Z3  Z2 g ¼ fu0 v1 : u 2 f32; 30; 0; 00g; f00; 01; 10; 11; 20; 21gg, as in Figure 1.

For all a 2 A; c 2 Zs and bd 2 Z2nþ1 the resulting symmetric starter graph has the following edges set:

Lemma 2.1. Let m, n, r, s be positive integers such that m = rs. Then there exists an ODC of Kmn ;mn by Krn ;sn .

[fðabÞ0 ðcdÞ1 : d ¼ 2n8b 2 f22n  3gg [ fðabÞ0 ðcdÞ1 : b ¼ 0 4 6 d 6 2n  3g

Proof. According to the symmetric starter vector III, v(Kr,s) is a symmetric starter vector with respect to Zm . Applying Theon times

rem (1.4) proves overbracevðKr;s Þ  vðKr;s Þ      vðKr;s Þ is a symmetric starter vector with respect to Znm . Moreover, the resulting symmetric starter graph has the following edges set   EðKrn ;sn : Þ ¼ u0 v1 : 8u 2 An ; v 2 Zns with A ¼ fx : x ¼ sðr  jÞ 1 6 j 6 rg. h Theorem 2.2. Let m, n, r, s be positive integers such that m = rs. Then there exists an ODC Kmnmn by Kr,ns. Proof. According to the symmetric starter vectors III and II, v(Kr,s) and v(K1,n) are symmetric starter vectors with respect to Zm and Zn respectively. Applying Theorem (1.4) proves v(Kr,s) · v(K1,n) is a symmetric starter vector with respect to Zm  Zn . Moreover, the resulting symmetric starter graph has the following edges set: EðKr;ns Þ ¼ fu0 v1 : 8u 2 A fcg; v 2 Zs  Zn g with A = {x:x = s(r  j), 1 6 j 6 r}. h In the next theorems we construct an ODC of Kmk,mk by G where G is disjoint union of complete bipartite graphs. Also we denote the vertices of G by (ab)0and(cd)1 the edges set of G by {(ab)0(cd)1} where ab and cd 2 Zm  Zk ; 8a 2 A; c 2 Zs and b; d 2 Zk . Then we can conclude the following three theorems for k; k = 2n + 1, k = 2n and k = n ” 1, 5 mod 6, where n is a positive integer number. Theorem 2.3. Let m, n,r,s be positive integers such that m = rs and n P 4. Then there exists an ODC of K2mn+m, 2mn+m by Kr,s [ Kr,4s [ Kr,2(n3)s [ K2r,s. Proof. According to the symmetric starter vectors III and V, (Kr,s) and vðK1;4 [ K2;1 [ K1;1 [ K1;2n6 Þ are symmetric starter vectors with respect to Zm and Z2nþ1 respectively. By Theorem (1.4), vðKr;s Þ  vðK1;4 [ K2;1 [ K1;1 [ K1;2n6 Þ is a symmetric starter vector with respect to Zm  Z2nþ1 .

EðKr;s [ Kr;4s [ Kr;2ðn3Þs [ K2r;s Þ ¼ fðabÞ0 ðcdÞ1 : b ¼ 2n; 8d 2 f012n  22n  1gg

[fðab0 ÞðcdÞ1 : b ¼ d ¼ 3g:

As special case if n = 5, we construct an ODC of K2mn+m, 2mn+m by Kr,s [ 2Kr,4s [ K2r,s. h Theorem 2.4. Let m, n, r, s be positive integers such that m = rs and n P 2. Then there exists an ODC of K2mn,2mn by 2Kr,sn. Proof. According to the symmetric starter vectors III and IV, v(Kr,s) and v(2K1,n) are symmetric starter vectors with respect to Zm and Z2n respectively. Applying Theorem (1.4) proves v(Kr,s) · v(2K1,n) is a symmetric starter vector with respect to Zm  Z2n . For all a 2 A; c 2 Zs and bd 2 Z2n we have 2 different cases to define the edges set of the resulting symmetric starter graph: Case 1: n is even Eð2Kr;sn Þ ¼ fðabÞ0 ðcdÞ1 : b ¼ n  2; d is oddg [ fðabÞ0 ðcdÞ1 : b ¼ n; d is eveng: Case 2: n is odd Eð2Kr;sn Þ ¼ fðabÞ0 ðcdÞ1 : b ¼ n  2; d is eveng [ fðabÞ0 ðcdÞ1 : b ¼ n; d is oddg:



Theorem 2.5. Let m, n, r, s n be positive integers such that m = rs and n ” 1, 5 mod6. Then there exists an ODC of Kmn,mn by nKr,s. Proof. According to the symmetric starter vectors III and I, v(Kr,s) and v(nK2) are symmetric starter with respect to Zm and Zn respectively. Applying Theorem (1.4) proves v(Kr,s) · v(nK2) is a symmetric starter vector with respect to Zm  Zn . Moreover, the resulting symmetric starter graph has the following edges set:

660 EðnKrs Þ ¼ fðabÞ0 ðcdÞ1 : 8b 2 Zn ; d ¼ 2b 2 Zn g: 

S. El-Serafi et al. a 2 A;

c 2 Zs and

3. Conclusion This paper is concerned with the orthogonal double cover of complete bipartite graphs by a complete bipartite graph and the disjoint union of complete bipartite graphs. The cartesian product of symmetric starter vectors is used to construct our results.

Ramadan Abd El-Hameed El-Shanawany Dr. Ramadan El-Shanawany received his M.Sc. in Pure Mathematics from University of Menoufiya, Egypt, he received his PhD. degree in Discrete Mathematics from university of Rostock, Germany. He is an associate professor at faculty of Electronic Engineering, Menoufiya University, Egypt. His area of interest is mutually orthogonal graph squares discrete mathematics and function analysis. He has published many research papers in various international journals and conferences.

References [1] Gronau H-DOF, Hartmann S, Gru¨ttmu¨ller M, Leck U, Leck V. On orthogonal double covers of graphs. Des Codes Cryptogrph 2002;27:49–91. [2] El Shanawany R, Higazy M, Scapellato R. Orthogonal double covers of complete bipartite graphs by the union of a cycle and a star. Aust J Comb 2009;43:281–93. [3] Balakrishnan R, Ranganathan K. Text book of graph theory. Berlin: Springer; 2012. [4] El-Shanawany R, Gronau H-DOF, Gru¨ttmu¨ller Martin. Orthogonal double covers of Kn,n by small graphs. Discrete Appl Math 2004;138:47–63. [5] El Shanawany R, Higazy M, Shabana H. Orthogonal double covers of a complete Bipartite graph by a complete Bipartite graph. In: International conference on mathematics, trends and development ICMTD12, organized by The Egyptian Mathematical Society, Cairo, Egypt; 27–29 December 2012. [6] El Shanawany R, Higazy M, El Mesady A. On Cartesian products of orthogonal double covers. Int J Math Math Sci 2013:265136. [7] El Shanawany R, Shabana H. Orthogonal double covers of complete bipartite graphs by a special class of disjoint union of path and a complete bipartite graph. Brit J Math Comput Sci 2013;3(3).

Said Ali Serafi Prof Dr Said Elserifi was born in Alexandria, Egypt. He received his M.Sc. and PhD. Degree from Alexandria University. He is a professor of Engineering Mathematics at faculty of Electronic Engineering, Menoufiya University, Egypt. He was the Vice Dean of the faculty of Electronic Engineering, Menoufiya University for Community and Environmental Development. His current interests are differential equations, numerical analysis and control systems. He has published many research papers in various international journals and conferences.

Hanan Magdy Shabana Mrs. Hanan Shabana Obtained her B.Sc. degree in Electronic Engineering from Menoufiya University and her M.Sc. degree in Physics and Engineering Mathematics from Menonfiya University, Egypt in 2014. She works as an assistant lecturer at Menoufiya University in Egypt. She has published many research papers in international journals and well-known international conferences. Her main research interest includes graph theory, Combinatorics and discrete mathematics.