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

TL;DR

This paper proves that every 2-connected $[4, 2]$-graph with at least seven vertices has an edge lying on cycles of all lengths from 3 up to the number of vertices, strengthening previous results.

Contribution

It establishes the existence of pancyclic edges in 2-connected $[4, 2]$-graphs of order at least seven and determines the minimum size of such graphs, also showing they are not uniquely Hamiltonian.

Findings

Every 2-connected $[4, 2]$-graph of order ≥7 contains a pancyclic edge.

Minimum size of $[4, 2]$-graphs of a given order is determined.

Any $[4, 2]$-graph of order ≥8 is not uniquely Hamiltonian.

Abstract

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ 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.