The kk-dominating cycles in graphs

For a graph GG, let σ̄k+3(G)=min{d(x1)+d(x2)+⋯+d(xk+3)−|N(x1)∩N(x2)∩⋯∩N(xk+3)|∣x1,x2,…,xk+3 are k+3 independent vertices in G}. In [H. Li, On cycles in 3-connected graphs, Graphs Combin. 16 (2000) 319–335], H. Li proved that if GG is a 3-connected graph of order nn and σ̄4(G)≥n+3, then GG has a maximum cycle such that each component of G−CG−C has at most one vertex. In this paper, we extend this result as follows. Let GG be a (k+2)(k+2)-connected graph of order nn. If σ̄k+3(G)≥n+k(k+2), GG has a cycle CC such that each component of G−CG−C has at most kk vertices. Moreover, the lower bound is sharp.