Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / R / 1 / i / Solution

False. A connected graph has an Eulerian cycle exactly when every vertex has even degree. In $K_n$ every degree is $n-1$, so it has an Eulerian cycle precisely when $n$ is odd. The complete graph $K_4$ is a counterexample to the statement for all $n\ge 3$.