Corrádi and Hajnal's Theorem for Sparse Random Graphs

Abstract

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that ifp(n) ≫ (logn/n)1/2, then asymptotically almost surely every subgraph ofG(n,p) with minimum degree at least (2/3 +o(1))npcontains a triangle packing that covers all but at mostO(p−2) vertices. Moreover, the assumption onpis optimal up to the (logn)1/2factor and the presence of the set ofO(p−2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.