The convex hull of degree sequences of signed graphs

As shown in [D. Hoffman, H. Jordon, Signed graph factors and degree sequences, J. Graph Theory 52 (2006) 27–36], the degree sequences of signed graphs can be characterized by a system of linear inequalities. The set of all nn-tuples satisfying this system of linear inequalities is a polytope PnPn. In this paper, we show that PnPn is the convex hull of the set of degree sequences of signed graphs of order nn. We also determine many properties of PnPn, including a characterization of its vertices. The convex hull of imbalance sequences of digraphs is also investigated using the characterization given in [D. Mubayi, T.G. Will, D.B. West, Realizing degree imbalances of directed graphs, Discrete Math. 239 (2001) 147–153].