Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n), and the circumference of a 3-connected claw-free graph is Ω(n). We generalise and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. Partially supported by NSF VIGRE Grant School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, England School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. Partially supported by an NSA grant and NSFC Project 10628102