Splitting an Expander Graph

Let G V E be an r regular expander graph Certain algorithms for nding edge disjoint paths require the edges of G to be partitioned into E E E Ek so that the graphs Gi V Ei are each expanders In this paper we give a non constructive proof of a very good split plus an algorithm which improves on that given in Broder Frieze and Upfal Existence and construction of edge disjoint paths on expander graphs SIAM Journal on Computing Introduction Let G V E be an r regular graph with jV j n For the asymptotics we shall assume that r is xed as n For S V let out S fe v w V v S w  Sg be the set of edges of G with exactly one endpoint in S Let S jout S jSj and let the edge expansion G of G be de ned by min S V jSj n S Loosely speaking G is an expander if is large There have been several papers recently which deal with the problem of joining selected pairs of vertices by edge disjoint paths In all of these papers we are given an expander graph and there is a need to partition the edges E E E Ek so that the graphs Gi V Ei i k are expanders A method was described in but it is relatively ine cient This computational problem seems intersting in its own right In this paper we prove two results One is non constructive and Department of Mathematical Sciences Carnegie Mellon Univer sity Pittsburgh PA USA Supported in part by NSF grant CCR E mail alan random math cmu edu Department of Computer Science University of Toronto Toronto Canada Supported in part by NSERC grant E mail molloy cs toronto edu shows what might be achieved The second is constructive The split is not as good but it does improve signi cantly on what is achieved in We use a subscript i to denote graph theoretic constructs related to Gi Thus di v is the degree of v in Gi Left unsubscripted such things refer to G Thus d v r In Section we prove Theorem Let k be a positive integer and let be a small positive real number Suppose that r log r k k log r Then there exists a partition E E E Ek such that for i k a i k b r k Gi Gi r k We have not been able to make the proof of this theorem constructive as in and Instead we will su ce ourselves with the following theorem proved in Section Assume that