Rank-width of random graphs

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour [J Combin Theory Ser B 96 (2006), 514-528]. We investigate the asymptotic behavior of rank-width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if pε(0, 1) is a constant, then rw(G(n, p)) = ⌈n/3⌉-O(1), (ii) if, then rw(G(n, p)) = ⌈1/3⌉-o(n), (iii) if p = c/n and c>1, then rw(G(n, p))≥rn for some r = r(c), and (iv) if p≤c/n and c81, then rw(G(n, p))≤2. As a corollary, we deduce that the tree-width of G(n, p) is linear in n whenever p = c/n for each c>1, answering a question of Gao [2006].