  We consider the following fundamental realization problem of directed graphs.
Given a sequence $S:={a_1 \choose b_1},\dots,{a_n \choose b_n}$ with
$a_i,b_i\in \mathbb{Z}_0^+.$ Does there exist a digraph (no loops and no
parallel arcs are allowed)$G=(V,A)$ with a labeled vertex set
$V:=\{v_1,\dots,v_n\}$ such that for all $v_i \in V$ indegree and outdegree of
$v_i$ match exactly the given numbers $a_i$ and $b_i$, respectively? There
exist two known approaches solving this problem in polynomial running time. One
first approach of Kleitman and Wang (1973) uses recursive algorithms to
construct digraph realizations \cite{KleitWang:73}. The second one draws back
into the Fifties and Sixties of the last century and gives a complete
characterization of digraph sequences (Gale 1957, Fulkerson 1960, Ryser 1957,
Chen 1966). That is, one has only to validate a certain number of inequalities.
Chen bounded this number by $n$. His characterization demands the property that
$S$ has to be in lexicographical order. We show that this condition is stronger
than necessary. We provide a new characterization which is formally analogous
to the classical one by Erd{\H o}s and Gallai (1960) for graphs. Hence, we can
give several, different sets of $n$ inequalities. We think that this stronger
result can be very important with respect to structural insights about the sets
of digraph sequences, for example in the context of threshold sequences.
Furthermore, the number of inequalities can be restricted to all $k \in
\{1,\dots,n-1\}$ with $a_{k+1}>a_{k}$ and to $k=n.$ An analogous result for
graphs was given by Tripathi and Vijay \cite{TripathiVijay03}. We prove this
property also for the case of digraphs (no parallel arcs) with at most one loop
per vertex.
