Linear Eulerian Extensions of Inhomogenous Random Graphs

The Eulerian extension number of any graph~$H$ (i.e. the minimum number of edges needed to be added to make~$H$ Eulerian) is at least~$t(H),$ half the number of odd degree vertices of~$H\mathrm{.}$ In this paper we consider an inhomogenous random graph~$G$ whose edge probabilities need not all be the same and use an iterative probabilistic method to obtain sufficient conditions for the Eulerian extension number of~$G$ to grow \emph{linearly} with~$t(G)\mathrm{.}$ We derive our conditions in terms of the average edge probabilities and edge density and also briefly illustrate our result with an example.