Minimal graphs for hamiltonian extension

TL;DR

This paper determines the minimum number of edges needed in an n-vertex graph to ensure that every non-edge can be added to form a Hamiltonian cycle, for all n ≥ 3.

Contribution

It establishes the exact minimal edge count for graphs with this Hamiltonian extension property, filling a gap in extremal graph theory.

Findings

Exact minimum edge counts for all n ≥ 3

Characterization of minimal graphs with Hamiltonian extension property

Extension of classical Hamiltonian cycle results

Abstract

For every $n\ge 3$ we determine the minimum number of edges of graph with $n$ vertices such that for any non edge $xy$ there exits a hamiltonian cycle containing $xy$.