Graphs without proper subgraphs of minimum degree 3 and short cycles

Abstract

We study graphs on n vertices which have 2n−2 edges and no proper induced subgraphs of minimum degree 3. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths 3,4,5,...,C(n) for some function C(n) tending to in finity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with n vertices and 2n−2 edges, containing no proper subgraph of minimum degree 3.

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Correspondence to Alexey Pokrovskiy.

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Research supported by the Research Training Group Methods for Discrete Structures and the Berlin Mathematical School.

Research supported by the Research Training Group Methods for Discrete Structures.

Research partially supported by DFG within the Research Training Group Methods for Discrete Structures.

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Narins, L., Pokrovskiy, A. & Szabó, T. Graphs without proper subgraphs of minimum degree 3 and short cycles. Combinatorica 37, 495–519 (2017). https://doi.org/10.1007/s00493-015-3310-9

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Mathematics Subject Classification (2000)

  • 05C55
  • 05C38