Graphs with Many Hamiltonian Paths

TL;DR

This paper constructs and analyzes families of graphs with many Hamiltonian paths, including non-hamiltonian graphs with ratios approaching 1 and minimal hamiltonian-connected graphs, advancing understanding of Hamiltonian properties.

Contribution

It introduces new constructions of non-hamiltonian graphs with high ratios of Hamiltonian paths and minimal hamiltonian-connected graphs matching known edge bounds.

Findings

Non-hamiltonian graphs with ratios approaching 1 for Hamiltonian paths

Infinite family of minimal hamiltonian-connected graphs with 3n/2 edges

New insights into Hamiltonian path connectivity in graphs

Abstract

A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio of pairs of vertices connected by hamiltonian paths to all pairs of vertices approaches 1. We then consider minimal graphs that are hamiltonian-connected. It is known that any order-$n$ graph that is hamiltonian-connected must have $\geq 3n/2$ edges. We construct an infinite family of graphs realizing this minimum.