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 subset of or equal to V(G) and \S\ less than or equal to 12, then either G has an Eulerian subgraph H such that S subset of or equal to V(H), 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 \S\ less than or equal to 23, G has an Eulerian subgraph H such that S subset of or equal to V(H). As an application, we obtain a new result on Hamiltonian line graphs. (C) 2003 Wiley Periodicals, Inc.