Signed graph factors and degree sequences

For a signed graph G and function $f: V(G) \rightarrow Z$ , a signed f ‐factor of G is a spanning subgraph F such that sdeg F ( υ ) =  f ( υ ) for every vertex υ of G , where sdeg( υ ) is the number of positive edges incident with v less the number of negative edges incident with υ , with loops counting twice in either case. For a given vertex‐function f , we provide necessary and sufficient conditions for a signed graph G to have a signed f ‐factor. As a consequence of this result, an Erdős‐Gallai‐type result is given for a sequence of integers to be the degree sequence of a signed r ‐graph, the graph with at most r positive and r negative edges between a given pair of distinct vertices. We discuss how the theory can be altered when mixed edges (i.e., edges with one positive and one negative end) are allowed, and how it applies to bidirected graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 27–36, 2006