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期刊论文
The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs
Science in China: Series A Mathematics 2006 Vol. 49, No. 3, 289-299,-0001,():
Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of Pv-factorization of Km,n. When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio’s conjecture is true when v = 4k − 1. In this paper we shall show that Ushio Conjecture is true when v = 4k +1, and then Ushio’s conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P4k+1-factorization of Km,n is (i) 2km ≤ (2k + 1)n,(ii) 2kn ≤ (2k +1)m, (iii) m+n ≡ 0 (mod 4k +1), (iv) (4k +1)mn/[4k(m+n)] is an integer.
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