Minimal Pancyclicity

TL;DR

This paper determines the minimum number of edges in pancyclic graphs for small n, provides exact values up to n=37, and explores the general behavior of this function, including a conjecture proof.

Contribution

It offers exact values of m(n) for n ≤ 37 and investigates the growth pattern of m(n), including a proof of the conjecture in specific cases.

Findings

Exact values of m(n) for n ≤ 37 obtained

m(n) is increasing for certain n, supporting the conjecture

Construction methods for pancyclic graphs up to 37 vertices

Abstract

A pancyclic graph is a simple graph containing a cycle of length $k$ for all $3\leq k\leq n$. Let $m(n)$ be the minimum number of edges of all pancyclic graphs on $n$ vertices. Exact values are given for $m(n)$ for $n\leq 37$, combining calculations from an exhaustive search on graphs with up to 29 vertices with a construction that works for up to 37 vertices. The behavior of $m(n)$ in general is also explored, including a proof of the conjecture that $m(n+1)>m(n)$ for all $n$ in some special cases.