Linear Eulerian Extensions of Inhomogenous Random Graphs

TL;DR

This paper investigates conditions under which the Eulerian extension number of inhomogeneous random graphs grows linearly with the number of odd degree vertices, using probabilistic methods based on edge probabilities and density.

Contribution

It introduces sufficient conditions for linear growth of the Eulerian extension number in inhomogeneous random graphs, extending previous results to more general probabilistic models.

Findings

Eulerian extension number can grow linearly with t(G) under certain conditions.

Derived conditions relate to average edge probabilities and graph density.

Illustrated results with a specific example.

Abstract

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~\(H.\) 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~\(t(G).\) We derive our conditions in terms of the average edge probabilities and edge density and also briefly illustrate our result with an example.