A Tight Upper Bound on Acquaintance Time of Graphs

In this note we confirm a conjecture raised by Benjamini et al. [BST13] on the acquaintance time of graphs, proving that for all graphs G with n vertices it holds that AC(G) = O(n3/2), which is tight up to a multiplicative constant. This is done by proving that for all graphs G with n vertices and maximal degree ∆ it holds that AC(G) ≤ 20∆n. Combining this with the bound AC(G) ≤ O(n2/∆) from [BST13] gives the foregoing uniform upper bound of all n-vertex graphs. We also prove that for the n-vertex path Pn it holds that AC(Pn) = n − 2. In addition we show that the barbell graph Bn consisting of two cliques of sizes dn/2e and bn/2c connected by a single edge also has AC(Bn) = n−2. This shows that it is possible to add Ω(n2) edges to Pn without changing the AC value of the graph.