Aharoni's rainbow cycle conjecture holds up to an additive constant

TL;DR

The paper proves an additive-constant version of Aharoni's rainbow-cycle conjecture for fixed $r$, showing a rainbow cycle of length at most $\frac{n}{r} + \alpha_r$ where $\alpha_r = O(r^5 \log^2 r)$. It additionally provides a defect version allowing some color classes to have size below $r$, with the same $\alpha_r$-type bound. The proof splits into "many non-stars" and "few non-stars" cases, employing a galaxy-based extension lemma, contraction techniques, and Shen-type bounds to close the induction on $n$. This advances the understanding of rainbow-cycle analogues of the CH conjecture and highlights directions for reducing the $r$-dependence toward a universal constant.

Abstract

In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $α_r \in O(r^5 \log^2 r)$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + α_r$.

Figures (1)

• Figure 1: An example of an edge-colored graph satisfying the hypotheses of Lemma \ref{['key-lemma']}. Unlabelled vertices are the neighbours of $\{u_1, \dots, u_7\}$ in $X$. Other vertices of $X$ and edges with both ends in $X$ are not depicted.

Theorems & Definitions (25)

• Conjecture 1.1: CH78
• Theorem 1.2
• Conjecture 1.3
• proof : Proof of \ref{['conj-ch']} assuming \ref{['conj-aharoni']}
• Theorem 1.4: DDFGGHMM21
• Theorem 1.5: CGHI22
• Theorem 1.6: HS22
• Theorem 1.7
• Corollary 1.8
• proof