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The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that, unlike the SDP relaxation that undergoes a phase transition in the logarithmic average degree regime, the LP relaxation fails in recovering the planted bisection with high probability in this regime. We show that the LP relaxation instead exhibits a transition from recovery to nonrecovery in the linear average degree regime. Finally, we present nonrecovery conditions for graphs with average degree strictly between linear and logarithmic. Funding: A. Del Pia is partially funded by Office of Naval Research (ONR) [Grant N00014-19-1-2322]. D. Kunisky is supported by the ONR [Grant N00014-20-1-2335], a Simons Investigator Award to Daniel Spielman, and the National Science Foundation [Grants DMS-1712730 and DMS-1719545]. [ABSTRACT FROM AUTHOR]