Bounds for the Twin-width of Graphs
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Bonnet, Kim, Thomassé, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an n-vertex graph is less than (n+√(nln n)+√(n)+2ln n)/2, and the twin-width of an m-edge graph is less than √(3m)+ m^1/4√(ln m) / (4· 3^1/4) + 3m^1/4 / 2. Conference graphs of order n (when such graphs exist) have twin-width at least (n-1)/2, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph G(n,p) with 1/n≤ p=p(n)≤ 1/2 is larger than 2p(1-p)n - (2√(2)+ε)√(p(1-p)nln n) asymptotically almost surely for any positive ε. Lastly, we calculate the twin-width of random graphs G(n,p) with p≤ c/n for a constant c<1, determining the thresholds at which the twin-width jumps from 0 to 1 and from 1 to 2.
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