恰有2个内度的2维Torus网络的定向图

设G是一个简单图且D是G的一个定向图.若对D中任意顶点x,d-(x)=a或b,则称G是a,b]可实现的.主要研究了2维Torus网络中a,b]可实现的充要条件.设H=Torus(p,k)是一个2维Torus网络,其中p和k是2个不小于3且奇偶性相同的正整数.设0≤a,b≤4,则H是a,b]可实现的当且仅当存在非负整数s和t使得s+t=kp且as+bt=2kp.

Orientations of 2 dimensional Torus networks with only two in-degrees

Let G be a simple graph and D be an orientation of G. Let a,b]be shorthand for the problem of realizing an orientation on G whose only in-degrees are a and b. This paper focuses on this problem in 2-dimensional Torus networks. Let H = Torus( p,k),where p,k≥3 and they have the same parity. Then a,b] is realizable if and only if there are non-negative integers s and t so that s + t = kp and as + bt = 2kp.