Proof of Koml\'os's conjecture on Hamiltonian subsets

TL;DR

This paper proves Komlós's conjecture for large degrees, showing that the complete graph minimizes Hamiltonian subsets among graphs with high minimum degree, and establishes a near-optimal bound for graphs with high average degree.

Contribution

It confirms Komlós's conjecture for large degrees and provides a stronger result quantifying the number of Hamiltonian subsets in graphs with high average degree.

Findings

Complete graph minimizes Hamiltonian subsets for large degrees

Graphs with high average degree have nearly twice as many Hamiltonian subsets as the complete graph

Identifies specific exceptions to the minimality condition

Abstract

Koml\'os conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.