Leaf-to-leaf paths and cycles in degree-critical graphs

Abstract

An $n$-vertex graph is degree 3-critical if it has $2n - 2$ edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szab\'o through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: - every $n$-vertex degree 3-critical graph has $\Omega(\log n)$ distinct cycle lengths; -every tree with maximum degree $\Delta \ge 3$ and $\ell$ leaves has at least $\log_{\Delta-1}\, ((\Delta-2)\ell)$ distinct leaf-to-leaf path lengths; - for every integer $N\geq 1$, there exist arbitrarily large 1-3 trees which have $O(N^{0.91})$ distinct leaf-to-leaf path lengths smaller than $N$, and, conversely, every 1-3 tree on at least $2^N$ vertices has $\Omega(N^{2/3})$ distinct leaf-to-leaf path lengths smaller than $N$. Several of our proofs rely on purely combinatorial means, while others exploit a connection to an additive problem that might be of independent interest.