A note on the acquaintance time of random graphs

William B. Kinnersley (1) , Dieter Mitsche (2) , Pawel Pralat (3)

1 Ryerson University [Toronto]
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Department of Mathematics

In this short note, we prove a conjecture of Benjamini, Shinkar and Tsur on the acquaintance time AC(G) of a random graph G \in G(n,p). It is shown that asymptotically almost surely AC(G)=O(log n/p) for G \in G(n,p), provided that pn-log n-log log n \to \infty (that is, above the threshold for Hamiltonicity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asymptotically almost surely K_n cannot be covered with o(log n/p) copies of a random graph G \in G(n,p), provided that np > n^{1/2+\epsilon} and p < 1-\epsilon for some \epsilon > 0. We conclude the paper with a small improvement on the general upper bound showing that for any n-vertex graph G, we have AC(G)=O(n^2/log n).

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

William B. Kinnersley, Dieter Mitsche, Pawel Pralat. A note on the acquaintance time of random graphs. The Electronic Journal of Combinatorics, 2013, 20 (3), pp.P52. ⟨hal-00923072⟩