2024/04/10 by George B. Mertzios, Mertzios, George B., Sotiris Nikoletseas +5
Computer Science · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Opportunistic and Delay-Tolerant Networks
paper · pdf · doi:10.48550/arxiv.2404.07147
openalex publication_date 2024/04/10 · openalex created_date 2024/04/12 · openalex updated_date 2026/07/28
We consider random simple temporal graphs in which every edge of the complete graph Kn appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant~δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately \frac2lognlog\frac1δ with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n2) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdos-Renyi random graphs model, the size of the maximum δ-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-TEMPORAL CLIQUE is unlikely to have very large probability of success.