完全二部多重图的K_2,4-因子分解

K_2,4-Factorization of Complete Bipartite Multigraphs

摘要: 如果完全二部多重图λK_m,n的边集可以划分为λK_m,n的K_p,q-因子，则称λK_m,n存在K_p,q-因子分解。当p = 1、q = 2和p = 2、q = 3时，λK_m,n的K_p,q-因子分解的存在性问题已被完全解决。当p = 1、q = 3和p = 1、q = 4时，K_m,n的K_p,q-因子分解的存在性问题已被基本解决。文章研究当p = 2和q = 4时完全二部多重图λK_m,n的K_2,4-因子分解的存在性。证明完全二部多重图λK_m,n存在K_2,4-因子分解的充分必要条件是：1) m≡n≡0 (mod 2)，2) m ≤ 2n，3) n ≤ 2m，4)，m+n≡0 (mod 6)，5) 3λm,n/[4（m+n）]是整数。

Abstract: Let λK_m,n be a complete bipartite multigraph with two partite sets having m and n vertices, re-spectively. A K_p,q -factorization λK_m,n is a set of edge-disjoint K_p,q -factors of λK_m,n . When p = 1, q = 2 and p = 2, q = 3, the K_p,q -factorization of λK_m,n has been completely solved. When p = 1, q = 3 and p = 1, q = 4, the K_p,q -factorization of K_m,n has been totally solved. In this article, the K_2,4-factorization of λK_m,n is researched. We will give a necessary and sufficient condition for K_2,4-factorization of λK_m,n, that is: 1) m≡n≡0 (mod 2), 2) m ≤ 2n, 3) n ≤ 2m, 4) m+n≡0 (mod 6), 5) 3λm,n/[4(m+n)] .