Abstract
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.
Original language | English |
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Pages (from-to) | 462-505 |
Number of pages | 44 |
Journal | Random Structures and Algorithms |
Volume | 60 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2022 |
Keywords
- blocks
- cycles
- planar graphs
- Pólya urn
- random graphs
ASJC Scopus subject areas
- Software
- Applied Mathematics
- General Mathematics
- Computer Graphics and Computer-Aided Design