A Note on Cycles Containing 4 Vertices and 2-Path in 3-Connected Cubic Graphs

Finding a cycle that contains certain prescribed vertices or edges within a graph is a classical problem in graph theory. It has been established that every [Formula: see text]-connected cubic graph contains a cycle through any nine vertices, which is referred to as the well-known nine-point theorem. In this paper, it is demonstrated that every [Formula: see text]-connected cubic graph contains a cycle through any four prescribed vertices and a [Formula: see text]-path. This result is optimal.