Every $2$-connected $[4,2]$-graph of order at least seven contains a pancyclic edge

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t\mathrm{.}$ An edge $e$ in a graph $G$ of order $n$ is called pancyclic if for every integer $k$ with $3\le k\le n,$ $e$ lies in a $k$-cycle. We prove that every $2$-connected $[4,2]$-graph of order at least seven contains a pancyclic edge. This strengthens an existing result. We also determine the minimum size of a $[4,2]$-graph of a given order and show that any $[4,2]$-graph of order at least eight is not uniquely hamiltonian.