AbstractWe show that a necessary and sufficient condition for the existence of an Sk-factorization of the symmetric complete bipartite multi-digraph λKm,n∗ is m=n≡0(modk(k−1)/d), where d=(λ,k−1)

Ushio, Kazuhiko

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Star-factorization of symmetric complete bipartite multi-digraphs 

Discrete Mathematics 215 (2000) 293{297
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Note
Star-factorization of symmetric complete
bipartite multi-digraphs
Kazuhiko Ushio 
Department of Industrial Engineering, Faculty of Science and Technology, Kinki University,
Osaka 577-8502, Japan
Received 25 February 1999; revised 30 June 1999; accepted 6 July 1999
Abstract
We show that a necessary and sucient condition for the existence of an Sk -factorization
of the symmetric complete bipartite multi-digraph Km;n is m = n 0 (mod k(k − 1)=d), where
d= (; k − 1). c© 2000 Elsevier Science B.V. All rights reserved.
Keywords: Star-factorization; Symmetric complete bipartite multi-digraph
1. Introduction
The symmetric complete bipartite multi-digraph Km;n is the symmetric complete
bipartite digraph Km;n in which every arc is taken  times. Let Sk (k>3) denote the ori-
entation of the star K1; k−1 in which all arcs are directed away from the center-vertex to
end-vertices. A spanning subgraph F of Km;n is called an Sk -factor if each component
of F is isomorphic to Sk . If Km;n is expressed as an arc-disjoint sum of Sk -factors,
then this sum is called an Sk -factorization of Km;n. In this paper, it is shown that a
necessary and sucient condition for the existence of such a factorization is m= n 
0 (mod k(k − 1)=d), where d= (; k − 1).
Let Kn1 ; n2 , K

n1 ; n2 , Kn1 ; n2 ; n3 , K

n1 ; n2 ; n3 , and K

n1 ; n2 ;:::; nm denote the complete bipartite graph,
the symmetric complete bipartite digraph, the complete tripartite graph, the symmetric
complete tripartite digraph, and the symmetric complete multipartite digraph, respec-
tively. Let C^k , S^k , P^k , and K^p;q denote the cycle or the directed cycle, the star or
the directed star, the path or the directed path, and the complete bipartite graph or
the complete bipartite digraph, respectively, on two partite sets Vi and Vj. Let Sk
 Tel.: +81-6-721-2332; fax: +81-6-730-1320.
E-mail address: ushio@is.kindai.ac.jp (K. Ushio)
0012-365X/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.
PII: S0012 -365X(99)00321 -0
294 K. Ushio /Discrete Mathematics 215 (2000) 293{297
and ~Sk denote the evenly partite star and semi-evenly partite star, respectively, on
three partite sets Vi, Vj1 , Vj2 . Then the problems of giving the necessary and su-
cient conditions of C^k -factorization of Kn1 ; n2 , K

n1 ; n2 , K

n1 ; n2 ; n3 , and K

n1 ; n2 ;:::; nm have been
completely solved by Enomoto et al. [3] and Ushio [10,13]. S^k -factorization of Kn1 ; n2 ,
Kn1 ; n2 , and K

n1 ; n2 ; n3 have been studied by Du [2], Martin [5,6], Ushio and Tsuruno [17],
Ushio [12], and Wang [18]. Ushio [9] gives the necessary and sucient condition of
S^k -factorization of Kn1 ; n2 . Ushio [14,15] gives the necessary and sucient conditions
of Sk -factorization and ~Sk -factorization of Kn1 ; n2 ; n3 . P^k -factorization of Kn1 ; n2 and K

n1 ; n2
have been studied by Ushio and Tsuruno [16], and Ushio [7,8]. K^p;q-factorization of
Kn1 ; n2 has been studied by Martin [5]. Ushio [11] gives the necessary and sucient
condition of K^p;q-factorization of Kn1 ; n2 . For graph theoretical terms, see [1,4].
2. Sk-factor of Km;n
The following theorem is on the existence of Sk -factors of Km;n.
Theorem 1. Km;n has an Sk -factor if and only if (i) m + n  0 (mod k);
(ii) (k − 1)n − m  0 (mod k(k − 2)); (iii) (k − 1)m − n  0 (mod k(k − 2));
(iv) m6(k − 1)n and (v) n6(k − 1)m.
Proof. Suppose that Km;n has an Sk -factor F . Let t be the number of components of
F . Then t= (m+ n)=k. Hence, Condition (i) is necessary. Among these t components,
let t1 and t2 be the number of components whose center-vertices are in V1 and V2,
respectively. Then, since F is a spanning subgraph of Km;n, we have t1+(k−1)t2=m
and (k−1)t1+ t2=n. Hence t1=((k−1)n−m)=k(k−2) and t2=((k−1)m−n)=k(k−2).
From 06t16m and 06t26n, we must have m6(k−1)n and n6(k−1)m. Conditions
(ii){(v) are, therefore, necessary.
For those parameters m and n satisfying (i){(v), let t1 = ((k − 1)n − m)=k(k − 2)
and t2 = ((k − 1)m− n)=k(k − 2). Then t1 and t2 are integers such that 06t16m and
06t26n. Hence, t1 + (k − 1)t2 = m and (k − 1)t1 + t2 = n. Using t1 vertices in V1
and (k − 1)t1 vertices in V2, consider t1 Sk ’s whose end-vertices are in V2. Using the
remaining (k − 1)t2 vertices in V1 and the remaining t2 vertices in V2, consider t2 Sk ’s
whose end-vertices are in V1. Then these t1 + t2 Sk ’s are arc-disjoint and they form an
Sk -factor of Km;n.
Corollary 2. Kn;n has an Sk -factor if and only if n  0 (mod k).
3. Sk-factorization of Km;n
We use the following notation.
K. Ushio /Discrete Mathematics 215 (2000) 293{297 295
Notation. Given an Sk -factorization of Km;n, let
r be the number of factors
t be the number of components of each factor
b be the total number of components.
Among t components of each factor, let t1 and t2 be the numbers of components
whose center-vertices are in V1 and V2, respectively.
Among r components having vertex x in Vi, let rij be the numbers of components
whose center-vertices are in Vj.
We give the following necessary condition for the existence of an Sk -factorization
of Km;n.
Theorem 3. Let d = (; k − 1). If Km;n has an Sk -factorization; then
m= n  0 (mod k(k − 1)=d).
Proof. Suppose that Km;n has an Sk -factorization. Then b=2mn=(k−1), t=(m+n)=k,
r=b=t=2kmn=(k−1)(m+n), t1=((k−1)n−m)=k(k−2), t2=((k−1)m−n)=k(k−2),
m6(k − 1)n, and n6(k − 1)m. Moreover, (k − 1)r11 = n, r12 = n, (k − 1)r22 = m,
and r21 = m. Thus we have r= r11 + r12 = kn=(k− 1) and r= r21 + r22 = km=(k− 1).
Therefore, m = n holds. Moreover, when m = n, we have b = 2n2=(k − 1), t = 2n=k,
r = kn=(k − 1), t1 = t2 = n=k, r11 = r22 = n=(k − 1), and r12 = r21 = n. Therefore,
n  0 (mod k(k − 1)=d) holds, too.
We prove the following extension theorems, which we use later in this paper.
Theorem 4. If Kn;n has an Sk -factorization; then sK

n;n has an Sk -factorization for
every positive integer s.
Proof. Obvious. Construct an Sk -factorization of Kn;n repeatly s times. Then we have
an Sk -factorization of sKn;n.
Theorem 5. If Kn;n has an Sk -factorization; then K

sn; sn has an Sk -factorization for
every positive integer s.
Proof. Sk can be denoted as K1; k−1. When Kn;n has an Sk -factorization, K

n;n has a
K1; k−1-factorization. Therefore, Ksn; sn has a Ks; (k−1)s-factorization. Obviously, Ks; (k−1)s
has an Sk -factorization. Therefore, Ksn; sn has an Sk -factorization.
We use the following notation for an Sk .
Notation. For an Sk whose center-vertex is u and end-vertices are v1; v2; : : : ; vk−1, we
denote (u; v1; v2; : : : ; vk−1).
296 K. Ushio /Discrete Mathematics 215 (2000) 293{297
We give the following sucient conditions for the existence of an Sk -factorization
of Kn;n.
Theorem 6. When k − 1 = p and n= p(p+ 1); Kn;n has an Sk -factorization.
Proof. Let V1 = f1; 2; : : : ; ng, V2 = f10; 20; : : : ; n0g. For i = 1; 2; : : : ; p + 1 and
j = 1; 2; : : : ; p+ 1, construct (p+ 1)2 Sk -factors Fij as follows:
Fij = f((A+ 1); (B+ p+ 1; B+ p+ 2; : : : ; B+ p+ p)0)
((A+ 2); (B+ p+ p+ 1; B+ p+ p+ 2; : : : ; B+ p+ 2p)0)
: : :
((A+ p); (B+ p+ (p− 1)p+ 1; B+ p+ (p− 1)p
+2; : : : ; B+ p+ p2)0)
((B+ 1)0; (A+ p+ 1; A+ p+ 2; : : : ; A+ p+ p))
((B+ 2)0; (A+ p+ p+ 1; A+ p+ p+ 2; : : : ; A+ p+ 2p))
: : :
((B+ p)0; (A+ p+ (p− 1)p+ 1; A+ p+ (p− 1)p
+2; : : : ; A+ p+ p2))g;
where A= (i− 1)p, B= (j− 1)p, and the additions are taken modulo n with residues
1; 2; : : : ; n. Then they comprise an Sk -factorization of Kn;n.
Theorem 7. Let d = (; k − 1). When n  0 (mod k(k − 1)=d); Kn;n has an
Sk -factorization.
Proof. Put =d, k−1=pd, (; p)=1, n=sk(k−1)=d, and N =k(k−1)=d. Then we
have n=p(pd+1)s and N =p(pd+1). By Theorem 6, dKN;N has an Sk -factorization.
Applying Theorems 4 and 5, Kn;n has an Sk -factorization.
We have the following main theorem.
Main Theorem. Let d = (; k − 1). Km;n has an Sk -factorization if and only if
m= n  0 (mod k(k − 1)=d).
References
[1] G. Chartrand, L. Lesniak, Graphs & Digraphs, 2nd Edition, Wadsworth, Belmont, CA, 1986.
[2] B. Du, K1;p2 -factorisation of complete bipartite graphs, Discrete Math. 187 (1998) 273{279.
[3] H. Enomoto, T. Miyamoto, K. Ushio, Ck -factorization of complete bipartite graphs, Graphs Combin. 4
(1988) 111{113.
[4] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1972.
[5] N. Martin, Complete bipartite factorisations by complete bipartite graphs, Discrete Math. 167=168 (1997)
461{480.
[6] N. Martin, Balanced bipartite graphs may be completely star-factored, J. Combin. Des. 5 (1997)
407{415.
[7] K. Ushio, P3-factorization of complete bipartite graphs, Discrete Math. 72 (1988) 361{366.
K. Ushio /Discrete Mathematics 215 (2000) 293{297 297
[8] K. Ushio, G-designs and related designs, Discrete Math. 116 (1993) 299{311.
[9] K. Ushio, Star-factorization of symmetric complete bipartite digraphs, Discrete Math. 167=168 (1997)
593{596.
[10] K. Ushio, C^k -factorization of symmetric complete bipartite and tripartite digraphs, J. Fac. Sci. Technol.
Kinki Univ. 33 (1997) 221{222.
[11] K. Ushio, K^p;q-factorization of symmetric complete bipartite digraphs, in: Y. Alavi, D.R. Lick,
A. Schwenk (Eds.), Graph Theory, Combinatorics, Algorithms and Applications, New Issues Press,
Kalamazoo, MI, 1999, pp. 823{826.
[12] K. Ushio, S^k -factorization of symmetric complete tripartite digraphs, Discrete Math. 197=198 (1999)
791{797.
[13] K. Ushio, Cycle-factorization of symmetric complete multipartite digraphs, Discrete Math. 199 (1999)
273{278.
[14] K. Ushio, Sk -factorization of symmetric complete tripartite digraphs, Discrete Math. 211 (2000)
281{286.
[15] K. Ushio, Semi-evenly partite star-factorization of symmetric complete tripartite digraphs, Australas. J.
Combin., 1999, to appear.
[16] K. Ushio, R. Tsuruno, P3-factorization of complete multipartite graphs, Graphs Combin. 5 (1989) 385{
387.
[17] K. Ushio, R. Tsuruno, Cyclic Sk -factorization of complete bipartite graphs, in: Y. Alavi, F.R.K. Chung,
R.L. Graham, U.F. Hsu (Eds.), Graph Theory, Combinatorics, Algorithms and Applications SIAM,
Philadelphia, PA, 1991, pp. 557{563.
[18] H. Wang, On K1;k -factorizations of a complete bipartite graph, Discrete Math. 126 (1994) 359{364.


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