Bound vertices of longest paths between two vertices in cubic graphs

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. Let H be a subgraph of a graph G. A vertex v of H is said to be H-bound if all the neighbors of v in G lie in H. Recently, Zhan has made the more general conjecture that in a k-connected graph, every longest path P between two vertices contains at least $k-1$ internal P-bound vertices. In this paper, we prove that Zhan's conjecture holds for $2$-connected cubic graphs. This conclusion generalizes a result of Thomassen [{ J. Combin. Theory Ser. B} 129 (2018) 148--157]. Furthermore, we prove that if the two vertices are adjacent, Zhan's conjecture holds for $3$-connected cubic graphs, from which we deduce that every longest cycle in a $3$-connected cubic graph has at least two chords. This strengthens a result of Thomassen [{ J. Combin. Theory Ser. B} 71 (1997) 211--214].