Degree sequences in complexes and hypergraphs

Abstract

Given an n n -complex K K and a vertex v v in K K , the n n -degree of v v is the number of n n -simplexes in K K containing v v . The set of all n n -degrees in a complex K K is called its n n -degree sequence when arranged in nonincreasing order. The question “Which sequences of integers are n n -degree sequences?” is answered in this paper. This is done by generalizing the iterative characterization for the 1 1 -dimensional (graphical) case due to V. Havel. A corollary to this general theorem yields the analogous generalization for k k -graphs. The characterization of P. Erdös and T. Gallai is discussed briefly.