A Tight Upper Bound on Acquaintance Time of Graphs

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 𝒜𝒞(G) = O(n^3/2) . This is done by proving that for all graphs G with n vertices and maximum degree it holds that 𝒜𝒞(G) ≤ 20 n . Combining this with the bound 𝒜𝒞(G) ≤ O(n^2/ ) 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 𝒜𝒞(P_n)=n-2 . In addition we show that the barbell graph B_n consisting of two cliques of sizes ⌈ n/2⌉ and ⌊ n/2⌋ connected by a single edge also has 𝒜𝒞(B_n) = n-2 . This shows that it is possible to add (n^2 ) edges a graph without changing its 𝒜𝒞 value.