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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References
B. Bollobás: Modern Graph Theory, Springer, (1998).
B. Bollobás and G. Brightwell: Long cycles in graphs with no subgraphs of minimal degree 3, Discrete Math. 75 (1989), 47–53.
P. Erdős, R. J. Faudree, A. Gyárfás and R. H. Schelp: Cycles in graphs without proper subgraphs of minimum degree 3, Ars Combin. 25(B) (1988), 159–201.
P. Erdős, R. J. Faudree, C. Rousseau and R. H. Schelp: Subgraphs of minimal degree k, Discrete Math., 85(1) (1990), 53–58.
G. Laman: On graphs and the rigidity of plane skeletal structures, J. Engrg. Math. 4 (1970), 331–340.
A. Lehman: A solution of the Shannon switching game, J. Soc. Indust. Appl. 12 (1964), 687–725.
C. Nash-Williams: Edge disjoint spanning trees of finite graphs, J. London Math. Soc. 36 (1961), 445–450.
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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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DOI: https://doi.org/10.1007/s00493-015-3310-9