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

• Published in: Journal of combinatorial theory. Series B Vol. 101; no. 4; pp. 214 - 236
• Main Authors: Bilinski, Mark, Jackson, Bill, Ma, Jie, Yu, Xingxing
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2011
• Subjects: Circumference; Claw-free graph; Cubic graph; Cyclability; Edge-splitting; Eulerian subgraph; Line graph; Ryjáček closure; Line graph; Edge-splitting; Cyclability; Cubic graph; Claw-free graph; Circumference; Eulerian subgraph; Ryjáček closure
• ISSN: 0095-8956; 1096-0902
• DOI: 10.1016/j.jctb.2011.02.009

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.