Monochromatic cycle partitions of $r$-edge-coloured graphs with high minimum degree

TL;DR

The paper resolves a core question on how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an n-vertex, r-edge-coloured graph with minimum degree at least (1-δ)n, for δ∈(0,1/2). It proves cp_r(δ) is sandwiched between Ω(r ⌈ r/log(1/δ) ⌉) and O(r log r ⌈ r/log(1/δ) ⌉), up to a logarithmic factor, via a combination of the multicolor Szemerédi regularity lemma, an absorption technique, and a novel hypergraph transversal framework. A key contribution is the introduction of connecting hubs and linked hub families to enable robust absorption and by product disprove a conjecture of Bal–DeBiasio on monochromatic tree covering. The results advance understanding of how density (via high minimum degree) interacts with edge colourings to limit the number of monochromatic spanning structures, with implications for both cycle partitions and tree covers in dense coloured graphs.

Abstract

A question posed independently by Letzter and Pokrovskiy asks: how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an $r$-edge-coloured graph, as a function of its minimum (uncoloured) degree? We resolve this problem up to a $(\log r)$-factor. Specifically, we prove that, for any $r \geq 2$ and $δ\in (0,1/2)$, any $n$-vertex $r$-edge-coloured graph $G$ with $δ(G) \geq (1- δ)n$ can be covered with $\mathcal{O}(r \log r \cdot \lceil r/\log(1/δ)\rceil)$ vertex-disjoint monochromatic cycles. We construct graphs that show this is tight up to the $(\log r)$-factor for all values of $r$ and $δ$, and along the way disprove a conjecture of Bal and DeBiasio about monochromatic tree covering.

Figures (3)

• Figure 1: A connecting hub with colour set $C = \{\textrm{red}, \textrm{green}, \textrm{blue}\}$. For each $c \in C$, every vertex in $S$ sends many $c$-coloured edges to $T_c$. Moreover, every vertex in $X$ either sends many $c$-coloured edges to $T_c$ for some$c \in C$, or it sends few edges to $S$. Finally, the dashed red edge in $T_\textrm{red}$ represents an edge of $\tilde{G}_\textrm{red}(T_\textrm{red}, B,m )$, which guarantees a large red coneighbourhood in $B$.
• Figure 2: A visual depiction of the construction of the monochromatic $x$--$y$ path in \ref{['lem:connecting_lemma']}, assuming for simplicity that $s = 3$.
• Figure 3: A barbell.

Theorems & Definitions (77)

• Theorem 1.2
• Conjecture 1.3: baldebiasio
• Theorem 1.4
• Lemma 2.1: Chernoff's inequality for the hypergeometric distribution
• Lemma 2.2: Degree form of the multicolour regularity lemma
• Lemma 2.5
• Lemma 2.6
• Definition 2.7: sublinear expander
• Theorem 2.8: ks1ks2
• Theorem 2.9: ks1ks2