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

Mark Bilinski, Bill Jackson, Jie Ma, Xingxing Yu
2011 Journal of combinatorial theory. Series B (Print)  
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
more » ... ound 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.
doi:10.1016/j.jctb.2011.02.009 fatcat:cvfcg7aaefgo5nxuotyeidbvii