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(n0.694 ), and the circumference of a 3-connected claw-free graph is Ω(n0.121 ). We generalise and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m0.753 ) edges. We use this result together with the Ryj´aˇcek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n0.753 ). 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. ∗ 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 I...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Mark Bilinski, Bill Jackson, Jie Ma, Xingxing Yu
• Keywords: 3-connected Claw-free Graphs, Eulerian Subgraphs, Georgia Institute, JCT 2011