The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs

Abstract

Let K _m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P _v -factorization of K _m,n is a set of edge-disjoint P _v -factors of K _m,n which partition the set of edges of K _m,n . When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of P _v -factorization of K _m,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 P _4k+1-factorization of K _m,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.