Longest and shortest cycles in random planar graphs

LetP(n,m)be a graph chosen uniformly at random fromthe class of all planar graphs on vertex set{1,...,n}withm=m(n)edges. We study the cycle and block structure ofP(n,m)whenm∼n∕2. More precisely, we determine theasymptotic order of the length of the longest and shortestcycle inP(n,m)in the critical range whenm=n∕2+o(n).In addition, we describe the block structure ofP(n,m)in theweakly supercritical regime whenn2∕3≪m−n∕2≪n.