Gale-Ryser型刻划定理的一个推广

A Generalization of the Gale-Ryser Type Characterization Theorem

设Pm= p1,…,Pm及Qn=q1,…,qn 是两个由非负整数构成的不增序列。如果存在一个简单X,Y-二部图使得X中的顶点的度分别为 且Y中的顶点的度分别为 ,那么称序列对 是二部可图的。如果二部可图且任何两个来自不同部集的顶点之间最多连有t条边,那么称是t-二部可图的。本文给出一个t-二部可图序列的刻划定理。事实上,该定理是Gale-Ryser型刻划定理的一个推广。

Let and be two non-increasing sequences of nonnegative integers. The pair is said to be bigraphic if there is a simple X,Y-bigraph such that the vertices of X have degrees and the vertices of Y have degrees . is said to be t-bigraphic if it is bigraphic and no two vertices from different partite sets are joined by more than t edges. In this paper, we give a characterization for to be t-bigraphic. In fact, it is a gen-eration of the Gale-Ryser type characterization theorem.