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.
Highlights
1.1 Triangle packings in subgraphs of random graphsLet H be a fixed graph on h vertices, let G be a graph on n vertices
G has an H-factor if n is divisible by h and G contains n/h vertex-disjoint copies of H
We extend the result of Huang, Lee, and Sudakov to the sparse random graph setting in the case H = K3
Summary
Let H be a fixed graph on h vertices, let G be a graph on n vertices. An arbitrary collection of vertex-disjoint copies of H in G is called an H-packing in G. G(n, p) has a subgraph G with δ(G) ≥ (2/3 − γ)np whose largest triangle packing covers no more than (1 − γ)n vertices (e.g., we may let G be the intersection of G(n, p) with the complete 3-partite graph with color classes of sizes (1 + γ)n/3, n/3, and (1 − γ)n/3) Even though it was proved in [18] that p n−2/3(log n)1/3 guarantees that G(n, p) a.a.s. has a triangle-factor, the lower bound on p in Theorem 1.3 cannot be relaxed by more than the (log n)1/2 factor as if p n−1/2, a.a.s. one can remove all triangles from G(n, p) by deleting only o(np) edges incident to every vertex. The presence of the exceptional set of Dp−2 is indispensable, see Proposition 4.6 and [17, Proposition 6.3]
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