The spanning subgraphs of eulerian graphs

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2_m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.