Deviation Estimates for Eulerian Edit Numbers of Random Graphs

TL;DR

This paper investigates the number of edge edits needed to transform a random graph into an Eulerian graph, providing deviation estimates and showing that about n/4 operations are sufficient with high probability.

Contribution

It offers the first deviation estimates for Eulerian edit numbers in random graphs, covering addition, deletion, and combined operations, with probabilistic bounds.

Findings

Approximately n/4 operations suffice with high probability

Deviation estimates are established for three types of Eulerian edit numbers

Results apply to random graphs G(n,p) with high probability

Abstract

Consider the random graph~\(G(n,p)\) obtained by allowing each edge in the complete graph on~\(n\) vertices to be present with probability~\(p\) independent of the other edges. In this paper, we study the minimum number of edge edit operations needed to convert~\(G(n,p)\) into an Eulerian graph. We obtain deviation estimates for three types Eulerian edit numbers based on whether we perform only edge additions or only edge deletions or a combination of both and show that with high probability, roughly~\(\frac{n}{4}\) operations suffice in all three cases.