AbstractWe give a new short proof of Koren’s characterization of graphic lists, extendedto multigraphs with bounded multiplicity p, called p-graphs. The Edge-Count Cri-terion (ECC) for an integer n-tuple d and integer p is the statement that for alldisjoint sets I and J of indices,Pi∈Idi +Pj∈J[p(n − 1) − dj] > p|I| |J|. An integerlist d is the degree list of a p-graph if and only if it has even sum and satisfies ECC.Analogous statements hold for bipartite or directed graphs, and an old character-ization of degree lists of signed graphs follows as a corollary of the extension tomultigraphs...