Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.41266

A note on decompositions into 3-chromatic graphs

Bertille Granet·Daniel Horsley·Jeffrey N. Santrio
View original source ↗Read source document ↗Export metadata ↓

Abstract

Delcourt and Postle recently introduced the `discharging in the dual' proof technique, leading to a breakthrough proof of Nash-Williams' conjecture on triangle decompositions. Independently, and almost simultaneously, the first and second authors built on the `condensation' method of Bryant, Dukes, Horsley, Maenhaut and Montgomery to determine the decomposition threshold of all odd cycles of length at least $73$. In this note, we explain how the `discharging in the dual' and `condensation' methods can be combined to determine the decomposition threshold of cycles of any odd length, as well as the approximate decomposition threshold of a range of other $3$-chromatic graphs.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bertille Granet, Daniel Horsley, Jeffrey N. Santrio. 2026-10-30. A note on decompositions into 3-chromatic graphs. https://arxiv.org/abs/2610.41266

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Connected Dominating Sets in Triangulations

A dominating set of a graph $G$ is \emph{connected} if it induces a connected graph in $G$. For planar triangulations, it has been known since 1990 that every $n$-vertex triangulation with $n\ge4$ admits a connected dominating set of size at most $n/2 - 1$, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every triangulation admits a spanning tree with at least $11n/21$ leaves. This result has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings. The connected dominating set results extend readily to show that an $n$-vertex triangulation of a surface of Euler genus $g$ has a connected dominating set of size at most $10n/21+O(\sqrt{gn})$.

math.CO↗
arXiv2026

Curves on the torus with few intersections

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.

math.CO↗
arXiv2026

Spectral extrema on closed surfaces of fixed Euler genus

We investigate a classical extremal problem in combinatorics and topological graph theory: determining the maximum spectral radius $spex(n,γ)$ of all $n$-vertex graphs embeddable on a closed surface of Euler genus $γ$. Building on seminal contributions by Boots and Royle, Cao and Vince, and Hong in the 1990s; Ellingham and Zha in 2000; Dvořák and Mohar in 2010; Tait and Tobin in 2017; and Liu, Ning, and Wang in 2026, we provide a complete characterization of spectral extremal graphs for any fixed Euler genus $γ$ and all sufficiently large $n$. The extremal graph attaining $\operatorname{spex}(n,γ)$ is unique, necessarily edge-extremal, and, roughly speaking, is constructed from the join $K_2 \nabla P_{n-2}$ by adding exactly $3γ$ edges. As a consequence, we derive asymptotically tight estimates for $\operatorname{spex}(n,γ)$, strengthening a classical bound due to Ellingham and Zha [J. Combin. Theory B, 2000]. We further characterize the extremal structures for the projective plane and torus, sharpening earlier bounds of Hong [J. Combin. Theory B, 1995]. For the planar case, we establish an explicit vertex threshold of $4.5\times 10^6$ for the Tait--Tobin theorem. Our approach combines extremal constructions of cellular embeddings, topological edge-switching techniques, and walk-counting estimates for large-scale substructures, revealing a bridge between spectral extremality and surface topology.

math.CO↗