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

TL;DR

This work advances the understanding of cycle-length richness in sparse degree-critical graphs and leaf-to-leaf path structures in 1-3 trees. It proves that every $n$-vertex degree-$3$-critical graph contains $\Omega(\log n)$ distinct cycle lengths, and that a tree with maximum degree $\Delta$ and $\ell$ leaves has at least $\log_{\Delta-1}((\Delta-2)\ell)$ leaf-to-leaf path lengths. It also provides near-optimal upper and lower bounds on short leaf-to-leaf lengths in 1-3 trees, including a construction achieving $O(N^{0.91})$ distinct short lengths and a matching $\Omega(N^{2/3})$ bound in the large-$n$ regime, with deep connections to additive combinatorics. The approaches combine purely combinatorial methods with an additive-structure viewpoint, using vine constructions, forward/backward path trees, and the Dilworth theorem to translate structural properties into cycle-length conclusions. Overall, the paper sharpens the landscape of cycle lengths in sparse graphs and leaf-to-leaf path lengths in trees, and opens several avenues at the intersection of graph theory and additive combinatorics.

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, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó 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 $Ω(\log n)$ distinct cycle lengths; -every tree with maximum degree $Δ\ge 3$ and $\ell$ leaves has at least $\log_{Δ-1}\, ((Δ-2)\ell)$ 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 $Ω(N^{2/3})$ 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.

Figures (2)

• Figure 1: The construction of the tree $T_n((a_i))$. Each subtree $T_{i+mj}$ for $j\geq 1$, except $T_t$, represents a perfect binary tree on $a_i$ layers, whose root neighbours the corresponding vertex on the horizontal path $P$. Note that this pattern repeats cyclically every $m$ steps. To the vertex $v_t$, we instead append the specific tree $T_t$, as described in the context.
• Figure 2: An example of the forward path $P$ and the path collection $\mathcal{Q}$ forming a vine. The cycle $C$ is illustrated by the bold line.

Theorems & Definitions (33)

• Conjecture A: NPSz
• Theorem 1
• Conjecture B: NPSz
• Theorem 2
• Conjecture C: NPSz
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof