Notes on Growing a Tree in a Graph

We study the height of a spanning tree T of a graph G obtained by starting with a single vertex of G and repeatedly selecting, uniformly at random, an edge of G with exactly one endpoint in T and adding this edge to T . ÀThis research was partly supported by NSERC and by NSF grants DMS1661063 and DMS1362785. Some of this work was carried out while Vida Dujmović, Pat Morin, and Bruce Reed were visiting IMPA in Rio de Janeiro. They thank the Institute for its hospitality. ÂSchool of Computer Science, McGill University, Montréal, Canada ÃDepartment of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ÊDepartment of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA ÄDepartment of Computer Science, University of British Columbia, Vancouver, Canada. This author was supported by an NSERC Postdoctoral Fellowship and a Simons-Berkeley Research Fellowship. Part of this work was done while he was visiting the Simons Institute for the Theory of Computing at UC Berkeley. ÅSchool of Computer Science, Carleton University, Ottawa, Canada ÆCNRS, Projet COATI, I3S (CNRS and UNS) UMR7271 and INRIA, Sophia Antipolis, France. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Brasil. Visiting Research Professor, ERATO Kawarabayashi Large Graph Project, Japan. i ar X iv :1 70 7. 00 08 3v 1 [ m at h. PR ] 1 J ul 2 01 7