A Note on the Characterization of Digraph Sequences

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.