EulerianGraphQ
EulerianGraphQ[g] yields True if the graph g is Eulerian (has an Eulerian circuit), and False otherwise.
Examples
Test if a complete graph is Eulerian:
EulerianGraphQ[CompleteGraph[5]]
(* True *)A cycle graph is always Eulerian:
EulerianGraphQ[CycleGraph[6]]
(* True *)Please visit the official Wolfram Language Reference for more details.