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 resultwhich extends another theoremof Catlin, in [P.A. Catlin, A reductionmethod 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. © 2011 Elsevier B.V. All rights reserved.