An Erdős-Pósa theorem for cycles and faces of distinct lengths

2026-07-07

AI Summary

This study addresses the existence of $k$ vertex-disjoint cycles of pairwise distinct lengths in a graph, or alternatively, whether deleting few vertices can reduce the number of distinct cycle lengths to at most $k-1$. It also investigates the presence of well-separated faces of distinct colors in graphs embedded on surfaces. By integrating tools from structural graph theory, graph embedding theory, radial graph constructions, and additive combinatorics, the work establishes the first Erdős–Pósa-type theorems for both distinct-length cycles and distinctly colored faces in embedded graphs, providing explicit bounds: $O(k^6 \operatorname{polylog} k)$ for cycles and $O(k^2 d g)$ for faces, where $d$ and $g$ relate to the embedding’s structure and genus. The results show that any graph either contains $k$ such cycles or admits a vertex set of size $O(k^6 \operatorname{polylog} k)$ whose removal limits cycle length diversity; analogous conclusions hold for surface-embedded graphs, with inherent barriers identified for further bound improvements.

Abstract

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $ψ$ on a surface $Σ$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(ψ)$ of $ψ$, and let $R(G,ψ)$ be the radial graph of $(G, ψ)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(ψ)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $ψ$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(ψ) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,ψ)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

Problem

• Erdős-Pósa theorem
• cycles of distinct lengths
• embedded graphs
• facial lengths
• vertex-disjoint cycles

Innovation

• Erdős–Pósa theorem
• distinct cycle lengths
• embedded graphs
• facial distances
• radial graph