Even Longer Cycles in Essentially 4-Connected Planar Graphs
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A planar graph is essentially 4-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least 2n+4/5, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least 5/8(n+2). This improves the previously best-known lower bound 3/5(n+2).
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