Deviation Estimates for Eulerian Edit Numbers of Random Graphs

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.