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

Authors:
Mark Bilinski;Bill Jackson;Jie Ma;Xingxing Yu
Affiliations:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA;School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, England, United Kingdom;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2011

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Abstract

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 @W(n^0^.^6^9^4), and the circumference of a 3-connected claw-free graph is @W(n^0^.^1^2^1). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with @W(m^0^.^7^5^3) edges. We use this result together with the Ryjacek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to @W(n^0^.^7^5^3). 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.