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 0.694 ) , and the circumference of a 3-connected claw-free graph is Ω ( n 0.121 ) . We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω ( m 0.753 ) 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 0.753 ) . 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.