Realizing degree sequences with k-edge-connected uniform hypergraphs

An integral sequence d = (d1, d2, . . . , dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r-uniform hypergraph with degree sequence d. Similarly, a sequence d is r-uniformmulti-hypergraphic if there is an r-uniformhypergraph (possibly with multiple edges) with degree sequence d. In this paper, it is proved that an r-uniform hypergraphic sequence d = (d1, d2, . . . , dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, . . . , n and n i=1 di ≥ r(n−1) r−1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs. It is also proved that a nonincreasing integral sequence d = (d1, d2, . . . , dn) is the degree sequence of a k-edge-connected r-uniform hypergraph (possibly with multiple edges) if and only if n i=1 di is a multiple of r , dn ≥ k and n i=1 di ≥ max{ r(n−1) r−1 , rd1}. © 2013 Elsevier B.V. All rights reserved.