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 [{\em J. Combin. Theory Ser. B} $\mathbf{\text{129}}$ (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 [{\em J. Combin. Theory Ser. B} $\mathbf{\text{71}}$ (1997) 211–214].