A note on the number of edges in a Hamiltonian graph with no repeated cycle length

TL;DR

This paper investigates the maximum number of edges in Hamiltonian graphs with no repeated cycle lengths, providing new constructions that surpass previous bounds for infinitely many values of n.

Contribution

It introduces a novel construction using difference sets in _n to achieve more edges in such graphs than previously known, for infinitely many n.

Findings

Constructs Hamiltonian graphs with more edges than the simple bound

Uses difference sets to ensure no repeated cycle lengths

Provides infinite families of graphs with optimal edge counts

Abstract

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such a graph can have at most $n + \sqrt{2n} +1 $ edges. Using difference sets in $\mathbb{Z}_n$, we show that for infinitely many $n$, there is an $n$-vertex Hamiltonian graph with $n + \sqrt{n - 3/4} - 3/2$ edges and no repeated cycle length.