Group connectivity of graphs satisfying the Chvatal-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) <= center dot center dot center dot <= d(n). The classical Chvatal ' 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 Chvatal-condition), then G is hamiltonian. Similarly, let G be a (simple) balanced bipartite graph on n >= 4 vertices and (d(1),..., dn) be the degree sequence of G with d(1) <= center dot center dot center dot <= dn. The classical Chvatal ' s theorem states that if d(m) >= m + 1 or dn/2 = n/2 - m + 1 for each m with 1 <= m <= n/4(called the Chvatal-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 Chvatal-condition, then G is A-connected if and only if G not equal C4, 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 Chvatal-condition, then G is A-connected if and only if G not equal C6. (c) 2023 Elsevier B.V. All rights reserved.