A Tight Upper Bound on Acquaintance Time of Graphs

Abstract

In this note we confirm a conjecture raised by Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) on the acquaintance time of graphs, proving that for all graphs G with n vertices it holds that $\mathcal {AC}(G) = O(n^{3/2})$. This is done by proving that for all graphs G with n vertices and maximum degree $\varDelta$ it holds that $\mathcal {AC}(G) \le 20 \varDelta n$. Combining this with the bound $\mathcal {AC}(G) \le O(n^2/\varDelta )$ from Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) gives the uniform upper bound of $O(n^{3/2})$ for all n-vertex graphs. This bound is tight up to a multiplicative constant. We also prove that for the n-vertex path $P_n$ it holds that $\mathcal {AC}(P_n)=n-2$. In addition we show that the barbell graph $B_n$ consisting of two cliques of sizes ${\lceil n/2\rceil }$ and ${\lfloor n/2\rfloor }$ connected by a single edge also has $\mathcal {AC}(B_n) = n-2$. This shows that it is possible to add $\varOmega (n^2$) edges a graph without changing its $\mathcal {AC}$ value.