2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224386We study a random even subgraph of a finite graph $G$ with a general edge-weight $p\in(0,1)$. We demonstrate how it may be obtained from a certain random-cluster measure on $G$, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of the $q=2$ random-cluster model on the dual lattice. The properties of such a graph are discussed, and are related to Schramm--Löwner evolutions (SLE).Version 2 includes material about random even graphs with general values of the edge-parameter p, together with a coupling-from-the-past algorithm for their simulation. Version 3 includes a treatment of infinite graphs, and is to appear in the Electronic Journal of CombinatoricsProbabilityMathematical Physics05C80, 60K35text