Many Hamiltonian subsets in large graphs with given density

TL;DR

This paper establishes a near optimal lower bound on the number of Hamiltonian subsets in large graphs with given density, improving previous extremal bounds by considering graph order and structure.

Contribution

It introduces a new lower bound for Hamiltonian subsets that accounts for graph order and structure, surpassing prior extremal bounds for various graph classes.

Findings

Provides a lower bound of approximately 2^{d+1} for Hamiltonian subsets.

Shows that C4-free graphs with minimum degree d have at least n*2^{d^{2-o(1)}} Hamiltonian subsets.

Improves understanding of Hamiltonian subsets in dense graphs with specific structural constraints.

Abstract

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ($\approx 2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graphs with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.