# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.06784