Three-regular Subgraphs of Four-regular Graphs

For any 4-regular graphG(possibly with multiple edges), we prove that, if the numberNof distinct Euler orientations ofGis such thatN ≢ 1 (mod 3), thenGhas a 3-regular subgraph. It gives the new 4-regular graphs with multiple edges which have no 3-regular subgraphs, for which we know the number of Euler orientations.