An implicit degree sum condition for cycles through specified vertices

In 1989, Zhu, Li and Deng introduced the concept of implicit degree. For a subset S of V(G), let iΔ2(S) denote the maximum value of the implicit degree sum of two vertices in S. In this paper, we prove that: Let G be a 2-connected graph on n vertices and X be a subset of V(G). If iΔ2(S)≥n for each independent set S of order κ(X)+1 in G[X], then G has a cycle containing all vertices of X. This result generalize one result of Yamashita (2008) [14].