Cycles within specified distance from each vertex Original Research Article

Let G be a graph and let f be a non-negative integer-valued function defined on V(G). Then a cycle C is called an f-dominating cycle if dG(v,C)⩽f(v) holds for each v∈V(G), where dG(v,C) denotes the distance between v and C. A set S is called an f-stable set if dG(u,v)⩾f(u)+f(v) holds for each pair of distinct vertices u, v in S, and denote by αf(G) the order of a largest f-stable set in G. In this paper, we prove that if a k-connected graph G (k⩾2) satisfies αf+1(G)⩽k, then G has an f-dominating cycle, where f+1 is the function defined by (f+1)(v)=f(v)+1. By taking an appropriate function as f, we can deduce a number of known results from this theorem.