Hamilton completion and the path cover number of sparse random graphs

TL;DR

This paper establishes that sparse random graphs can be covered by a near-optimal number of vertex-disjoint paths, and relates this to the minimal edge addition needed for Hamiltonicity, providing tight probabilistic bounds.

Contribution

It proves a tight bound on the path cover number in sparse random graphs and links it to the minimal edges required for Hamiltonian completion.

Findings

High probability, the path cover number is approximately (1/2) c e^{-c} n.

Adding at most this many edges makes the graph Hamiltonian.

The bounds are essentially tight for large c.

Abstract

We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is essentially tight. This is equivalent to showing that, with high probability, at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ edges can be added to $G$ to create a Hamiltonian graph.