Realizing degree sequences with -edge-connected uniform hypergraphs

An integral sequence d = ( d 1 , d 2 , … , d n ) 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 -uniform multi-hypergraphic if there is an r -uniform hypergraph (possibly with multiple edges) with degree sequence d . In this paper, it is proved that an r -uniform hypergraphic sequence d = ( d 1 , d 2 , … , d n ) has a k -edge-connected realization if and only if both d i ≥ k for i = 1 , 2 , … , n and ∑ i = 1 n d i ≥ 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 = ( d 1 , d 2 , … , d n ) is the degree sequence of a k -edge-connected r -uniform hypergraph (possibly with multiple edges) if and only if ∑ i = 1 n d i is a multiple of r , d n ≥ k and ∑ i = 1 n d i ≥ max { r ( n − 1 ) r − 1 , r d 1 } .