Graphs without proper subgraphs of minimum degree 3 and short cycles

@article{Narins2017GraphsWP,
title={Graphs without proper subgraphs of minimum degree 3 and short cycles},
author={Lothar Narins and A. Pokrovskiy and Tibor Szab{\'o}},
journal={Combinatorica},
year={2017},
volume={37},
pages={495-519}
}

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.

Every degree 3-critical graph on n vertices contains cycles of at least 3 log 2 n + O(1) distinct lengths. A similar conjecture could be made about leaf-leaf paths in trees

Conjecture 6.2

Problem 6.1. Is there a function C(n) tending to infinity such that every degree 3-critical graph on n vertices contains cycles of all lengths 4

Problem 6.1. Is there a function C(n) tending to infinity such that every degree 3-critical graph on n vertices contains cycles of all lengths 4