Let $G=(V,E)$ (Such that $\vert V\vert$) be a Hamiltonian cubic graph and $v\in V$. We represent a graph as a cycle with vertices labeled in such a way that $(0,\ldots ,n-1)$ is a Hamiltonian cycle and a set of edges between non-successive vertices of a cycle. Then let us define $L$ as a set of successive vertices in Ham. cycle labeled $v,v+1,\ldots,v-k-1$, where $\vert V \vert \ge k \ge 2$ (we consider every $i \in \mathbb{N}$ mod n).
Then my question is the following: why the size of $L$ is always $\vert V \vert -k$? And why any vertex of $L$ has at least two neighbors in $L$?