Kazuhiko Ushio

P3-Factorization of complete bipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of a P 3 -factorization of K m,n is (i) m + n =0 (mod 3), (ii) m ⩽ 2n, (iii) n ⩽ 2m and (iv) 3 mn/2(m + n ) is an integar.

G -designs and related designs

Abstract This is a survey on the existence of G -designs, bipartite G -designs and multipartite G -designs.

P 3 -factorization of complete bipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of a P 3 -factorization of K m,n is (i) m + n =0 (mod 3), (ii) m ⩽ 2n, (iii) n ⩽ 2m and (iv) 3 mn/2(m + n ) is an integar.

P3-factorization of complete multipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aP3-factorization ofKmn is (i)mn ≡ 0(mod 3) and (ii) (m − 1)n ≡ 0(mod 4).

C k -factorization of complete bipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aCk-factorization ofKm,n is (i)m = n ≡ 0 (mod 2), (ii)k ≡ 0 (mod 2),k ≥ 4 and (iii) 2n ≡ 0 (modk) with precisely one exception, namely m =n = k = 6.

Star-factorization of symmetric complete bipartite multi-digraphs

We show that a necessary and sufficient condition for the existence of an Sk-factorization of the symmetric complete bipartite digraph K * , is m = n -~ 0 (mod k(k 1)).

C k -factorization of symmetric complete bipartite and tripartite multi-digraphs

Abstract We show that a necessary and sufficient condition for the existence of a C k -factorization of the symmetric complete bipartite multi-digraph λK n 1 , n 2 ∗ is (i) k≡0 ( mod 2) and (ii) n 1 =n 2 ≡0 ( mod k/2) . We also show that a necessary and sufficient condition for the existence of a C k -factorization of the symmetric complete tripartite multi-digraph λK n 1 ,n 2 , n 3 ∗ is (i) k≡0 ( mod 2) and (ii) n 1 =n 2 =n 3 ≡0 ( mod k) .

S k -factorization of symmetric complete tripartite digraphs

Let Kn1,n2,n3∗ denote the symmetric complete tripartite digraph with partite sets V1,V2,V3 of n1,n2,n3 vertices each, and let Kp,q denote the complete bipartite digraph in which all arcs are directed away from p start-vertices in Vi to q end-vertices in Vj with {i,j}⊂{1,2,3}. We show that a necessary condition for the existence of a Kp,q-factorization of Kn1,n2,n3∗ is n1=n2=n3≡0(moddp′q′(p′+q′)) for p′+q′≡1,2(mod3) and n1=n2=n3≡0(moddp′q′(p′+q′)/3),2n1⩾pp′,2n1⩾qq′ for p′+q′≡0(mod3), where d=(p,q),p′=p/d,q′=q/d. Several sufficient conditions are also given.

Evenly partite star-factorization of symmetric complete tripartite multi-digraphs

Abstract We show that a necessary and sufficient condition for the existence of an S 2q+1 - factorization of the symmetric complete tripartite multi-digraph λK n 1 , n 2 , n 3 ∗ is (i) n 1 = n 2 = n 3 for q = 1 and (ii) n 1 = n 2 = n 3 = 0 (mod (2 q +1) q / d ) for q ≥ 2, where d = ( λ , q ).

Cycle-factorization of symmetric complete multipartite digraphs

Abstract First, we show that a necessary and sufficient condition for the existence of a C 3 -factorization of the symmetric tripartite digraph K n 1 ,n 2 ,n 3 ∗ , is n 1 = n 2 = n 3 . Next, we show that a necessary and sufficient condition for the existence of a C 2k -factorization of the symmetric complete multipartite digraph K n 1 , n 2 ,…, n m is n 1 = n 2 = … = n m = 0 (mod k ) for even m and n 1 = n 2 = … = ≡ 0 (mod 2 k ) for odd m .