Group connectivity of graphs satisfying the Chvátal-condition.

Let G be a (simple) graph on n ≥ 3 vertices and (d 1 , ... , d n) be the degree sequence of G with d 1 ≤ ⋯ ≤ d n . The classical Chvátal's theorem states that if d m ≥ m + 1 or d n − m ≥ n − m for each m with 1 ≤ m < n 2 (called the Chvátal-condition), then G is hamiltonian. Similarly, let G be a (simple) balanced bipartite graph on n ≥ 4 vertices and (d 1 , ... , d n) be the degree sequence of G with d 1 ≤ ⋯ ≤ d n . The classical Chvátal's theorem states that if d m ≥ m + 1 or d n 2 ≥ n 2 − m + 1 for each m with 1 ≤ m ≤ n 4 (called the Chvátal-condition), then G is hamiltonian. In this paper, for an abelian group A of order at least 4, we show that if a graph G satisfies the Chvátal-condition, then G is A -connected if and only if G ≠ C 4 , where C ℓ is a cycle of length ℓ. Moreover, for an abelian group A of order at least 5, we also show that if a balanced bipartite graph G satisfies the Chvátal-condition, then G is A -connected if and only if G ≠ C 6 . [ABSTRACT FROM AUTHOR]