Spanning subgraph with Eulerian components

A graph is k -supereulerian if it has a spanning even subgraph with at most k components. We show that if G is a connected graph and G ′ is the (collapsible) reduction of G , then G is k -supereulerian if and only if G ′ is k -supereulerian. This extends Catlin’s reduction theorem in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. For a graph G , let F ( G ) be the minimum number of edges whose addition to G create a spanning supergraph containing two edge-disjoint spanning trees. We prove that if G is a connected graph with F ( G ) ≤ k , where k is a positive integer, then either G is k -supereulerian or G can be contracted to a tree of order k + 1 . This is a best possible result which extends another theorem of Catlin, in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. Finally, we use these results to give a sufficient condition on the minimum degree for a graph G to be k -supereulerian.