The smallest degree sum that yields graphic sequences with a -connected realization

A non-increasing sequence π = ( d 1 , d 2 , … , d n ) of non-negative integers is said to be graphic if it is the degree sequence of a simple graph G on n vertices. Let A be an (additive) Abelian group. An extremal problem for a graphic sequence to have an A -connected realization is considered as follows: determine the smallest even integer σ ( A , n ) such that each graphic sequence π = ( d 1 , d 2 , … , d n ) with d n ≥ 2 and σ ( π ) = d 1 + d 2 + ⋯ + d n ≥ σ ( A , n ) has an A -connected realization. In this paper, we determine σ ( Z 3 , n ) for n ≥ 5 .