Eulerian subgraphs in 3-edge-connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3-edge-connected graph with S V ðGÞ and jSj 12, then eitherG has an Eulerian subgraphH such that S V ðHÞ, or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3-edge-connected planar graph, then for any jSj 23, G has an Eulerian subgraph H such that S V ðHÞ. As an application, we obtain a new result on Hamiltonian line graphs. 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003