Arc-disjoint arborescences of digraphs

The purpose of this paper is to give a necessary and sufficient condition for a digraph G to contain k arcdisjoint arborescences so that the number rooted at each vertex x of G lies in some prescribed interval which depends on x. A digraph G = (V, E) consists of a vertex set V and an arc set E such that each arc has its head and tail in V. Multiple arcs are allowed, but no loops. In this paper we consider only the finite digraphs of order 3 2 . For X V, let x = V X and let p ( X ) denote the number of arcs of G having their heads in X and tails in x. An arborescence of G is defined as a spanning tree directed in such a way that each vertex of G, except one called the root of the arborescence, is the head of exactly one arc of the tree. Z+ denotes the set of non-negative integers. Let f and g be given functions: V + Z'. For convenience, we write f ( X ) = C x E X f ( x ) and g ( X ) = CxEX g(x) for any X C V, and set f(4) = g(4) = 0. Most graphical terms and notations used in this paper can be found in [l]. The present paper is a natural continuation of [ 5 ] . Its purpose is to generalize the following theorem of Edmonds, which will be used in the proof of the main theorem of this paper. Theorem 1 [2]. Let f ( x ) be a given function: V + Z' and k = f(V). A digraph G = (V, E) contains k arc-disjoint arborescences, exactly f ( x ) of which are rooted at x for each x E V, if and only if for every X such that 4 # X V , p ( W 3fC3. (1) Journal of Graph Theory, Vol. 7 (1983) 235-240