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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  p ∈(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  c 81, 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]. © 2011 Wiley Periodicals, Inc. J Graph Theory.

Rank‐width of random graphs